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ELEMENTS 



ANALYTICAL MECHANICS, 



.iY\A/ 



Wr Bv C. BAETLETT, L L. D., 

rrcOFESsoR of natural and experimental philosophy in the united 

STATES military ACADEMY AT WEST POINT, 

AND 

AUTHOR OF ELEMENTS OF MECHANICS, ACOUSTICS AND OPTICS. 



THIRD EDITION, REVISED AND CORRECTED. 



NEW YOEK: 
PUBLISHED BY A. S. BARNES & COMPANY, 

No. 51 JOHN-STREET. 
CINCINNATI: H. W. DERBY <fc CO* 

1855. 




D 



6 



<^^<,^ 



Entered according to Act of Congress, in the year One Thousand 

Eight Hundred and Fifty-three, 

By W. H. C. BART LETT, 

In the Clerk's Office of the District Court of the United States for the Southern 

District of New -York. 



1 






J O N E S & D E N Y S E , G . W . W O I) , 

Stcreot.ypcrs, Printer, 

183 William-street. John-street, cor. Dutch. 



T 



PREFACE. 



The following pages were mainly prepared several years ago 
for the use of tlie author's class in the United States Military 
Academy. Their publication has been unavoidably postponed 
to the present time, and they are now offered to the public 
in the hope that they may contribute something to lighten 
the labor which every student must encounter at the threshold 
of the subject of which it is their purpose to treat. 

In accordance with the suggestions of much experience in 
the business of teaching, all unnecessary divisions and sub- 
divisions have been avoided. Tliey too often divert the mind 
from what is essential to that which is merely accidental, and 
prevent the formation of those habits of generalization wliich 
alone can give facility in acquiring and confidence in apply- 
ing any branch of knowledge. 

Mechanics has for its object to investigate the action of 
forces upon the various forms of bodies. All physical phe- 
nomena are but the necessary results of a perpetual conflict 
of equal and opposing forces, and the mathematical formula 
expressive of the laws of this convict must involve the whole 
doctrine of Mechanics. Tlie study of Mechanics should, there- 
fore, be made to consist simply in the discussion of this for- 



iv PREFACE. 

mula, and in it should be sought the explanation of all effects 
that arise from the action of forces. 

The principle of classification adopted, is that suggested by 
differences in the physical constitution of bodies, and, accord- 
ingly, the subject has been treated under the heads Mechanics 
OF Solids and Mechanics of FLriDS. Much time and space 
are thus saved, the attention of the student is kept con- 
stantly upon his subject, and the discussion divested to the 
utmost of all specialties. 



CONTENTS. 



INTRODUCTION. 

Preliminary Definitions 11 

Physics of Ponderable Bodies 14 

Primary Properties of Bodies , . . . 15 

Secondary Properties 16 

Force 20 

Physical Constitution of Bodies 22 

PAKT I. 

MECHANICS OF SOLIDS. 

Space, Time, Motion and Force 31 

Work 38 

Varied Motion 42 

Equilibrium 46 

The Cord 41 

The Muffle 48 

Equilibrium of a Rigid System — Virtual Velocities 50 

Principle of D'Alembert 55 

Free Motion 58 

Composition and Resolution of Oblique Forces 62 ' 

Composition and Resolution of Parallel Forces 75 



VI CONTENTS. 

PAGE. 

Work of Resultant and of Component Forces 82 

Moments 84 

Composition and Eesolution of Moments 88 

Translation of General Equations 91 

Centre of Gravity , 93 

Centre of Gravity of Lines 91 

Centre of Gravity of Surfaces 102 

Centre of Gravity of Volumes 109 

Centrobaryc Method 114 

Centre of Inertia 116 

Motion of the Centre of Inertia 118 

Motion of Translation 120 

General Theorem of Work and Living Force 120 

Central Forces 121 

Stable and Unstable Equilibrium 123 

Initial Conditions, Direct and Reverse Problem 126 

Vertical Motion of Heavy Bodies 12Y 

Projectiles 135 

Laws of Central Forces 149 

Rotary Motion 165 

Moment of Inertia, Centre and Radius of Gyration 1*75 

Impulsive Forces 185 

Motion under the Action of Impulsive Forces 181 

Motion of the Centre of Inertia 187 

Motion about the Centre of Inertia 189 

Angular Velocity 190 

Motion of a System of Bodies 195 

Motion of Centre of Inertia of a System 196 

Conservation of the Motion of the Centre of Inertia of a System 197 

Conservation of Areas 199 

Invariable Plane 201 

Pi-inciple of Living Force , 202 

System of the World 208 

Impact of Bodies 211 

Consti'aiued Motion on a Surface 218 

Constrained Motion on a Curve 220 

Constrained Motion about a Fixed Point 246 

Constrained Motion about a Fixed Axis 247 

Compound Perdulum 249 

Motion of a Body about an Axis imder the Action of Impulsive Fqj'ccs 258 

Balistic Pendulum , 259 



coNTEi^'TS, vii 
PAET II. 

MECHANICS OF FLUIDS. . 

PAGE. 

Introductory Remarks 26S 

^Mariotte's Law 265 

Law of Pressure, Density and Temperature 266 

Equal Trausmission of Pressure 268 

Motion of Fluid Particles 2^0 

Equilibrium of Fluids 280 

Pressure of Heavy Fluids 289 

Equilibrium and Stability of Floating Bodies 295 

Specific Gravity 304 

Atmospheric Pressure 316 

Bai'ometer 31^7 

Motion of Heavy Licompressible Fluids in Vessels 326 

Motion of Elastic Fluids in Vessels 338 

PAET III. 

APPLICATIOXS TO SIMPLE MACHINES, PUMPS, &c. 

General Principles of all Machines 345 

Friction 347 

Stiffness of Cordage 355 

Friction on Pivots 360 

Friction on Trunnions 365 

The Cord as a Simple Machine 369 

The Catenary 379 

Friction between Cords and Cylindrical Solids 381 

Inclined Plane »,.... 383 

The Lever 386 

Wheel and Axle 389 

Fixed Pulley 391 

Movable Pulley 394 

The Wedge 400 

The Screw 404 

Pumps 409 

The Siphon 419 

The Air-pump 421 



vm 



CONTENTS. 



TABLES. 

PAGB. 

Table I. — The Tenacities of Diflferent Substances, and the Resistances which 

they oppose to Direct Compression 428 

Table II. — Of the Densities and Volumes of "Water at Different Degrees of 
Heat, (according to Stampfer), for every 2? Degrees of Fahren- 
heit's Scale 430 

Table III. — Of the Specific Gravities of some of the most Important Bodies. . 431 

Table IV. — Table for Finding Altitudes ■with the Barometer 434 

Table V. — Co-efl5cient Values, for the Discharge of Fluids through thin 
Plates, the Orifices being Remote from the Lateral Faces of the 

Vessel 436 

Table VI. — Experiments on Friction, without Unguents. By M. Morin 43*7 

Table VII. — Experiments on Friction of Unctuous Surfaces. By M. Morin 440 

Table VIII. — Experiments on Friction with Unguents interposed. By M. Morin . 441 
Table X. — Of Weights necessary to Bend different Ropes around a Wheel one 

Foot in Diameter 445 

Table IX. — Friction of Trunnions in their Boxes 444 



The Greek Alphabet is here inserted to aid those who are not already familiar with 



it, in reading the 


parts of the text in which its letters occur. 




Letters. 


Names. 


Letters. 


Names 


A 


a 


Alpha 


N V 


Nu 


B 


/3e 


Beta 


H i 


Xi 


r 


yr 


Gamma 


O 


micron 


A 


5 


Delta 


11 ^-TT 


Pi 


E 


s 


Epsilon 


p p^ 


Rho 


Z 


?c 


Zeta 


2 dg 


Sigma 


H 


y] 


i;ta 


T t7 


Tau 


© 


^& 


Theta 


T u 


Upsilon 


I 


I 


Iota 


9 


Phi 


K 


X 


Kappa 


XX 


Chi 


A 


X 


Lambda 


Y + 


Psi 


M 


f* 


Mu 


n 0) 


Omega 



ELEMENTS 



ANALYTICAL MECHANICS. 



IlSrTEODUCTIO]^. 

The term nature is employed to signify the assemblage of all 
the bodies of the universe ; it includes whatever exists and 
is the subject of change. Of the existence of bodies we are 
rendered conscious by the impressions they make on our senses. 
Their condition is subject to a variety of changes, whence we 
infer that external causes are in operation to produce them ; and 
to investigate nature with reference to these changes and their 
causes, is the object of Physical Science. 

All bodies may be distributed into three classes, viz : unorgan- 
ized or inanimate^ organized or animated^ and the heavenly l)odies 
ov prvmary organizations. 

The unorganized or inanimate bodies, as mmerals, water, air, 
form the lowest class, and are, so to speak, the substratum for the 
others. These bodies are acted on solely by causes external to 
themselves ; they have no definite or periodical duration ; nothing 
that can properly be termed life. 

The organized or animated bodies, are more or less perfect 
individuals, possessing organs adapted to the performance of cer- 
tain appropriate functions. In consequence of an innate principle 



12 ELEMENTS OF ANALYTICAL MECHANICS. 

peculiar to them, known as vitality^ bodies of this class are con- 
stantly appropriating to themselves unorganized matter, changing 
its properties, and deriving, by means of this process, an increase 
of bulk. They also possess the faculty of reproduction. They 
retain only for a limited time the vital principle, and, when life 
is extinct, they sink into the class of inanimate bodies. The 
animal and vegetable kingdoms include all the species of this 
class on our earth. 

The celestial lodies^ as the fixed stars, the sun, the comets, 
planets and their secondaries, are the gigantic individuals of the 
universe, endowed with an organization on the grandest scale. 
Their constituent parts may be compared to the organs possessed 
by bodies of the second class ; those of our earth are its conti- 
nents, its ocean, its atmosphere, which are constantly exerting a 
vigorous action on each other, and bringing about changes the 
most important. 

The earth supports and nourishes both the vegetable and animal 
world, and the researches of Geology have demonstrated, that 
there was once a time when neither plants nor animals existed on 
its surface, and that prior to the creation of either of these orders, 
great changes must have taken place in its constitution. As the 
earth existed thus anterior to the organized beings upon it, we 
may infer that the other heavenly bodies, in like manner, were 
called into being before any of the organized bodies which pro- 
bably exist upon them. Eeasoning, then, by analogy from our 
earth, we may venture to regard the heavenly bodies as the pri- 
mary organized forms, on whose surface both animals and vege- 
tables find a place and support. 

JSfat'ural Philosophy^ or Physics^ treats of the general proper- 
ties of unorganized\)odi\Q?>^ of tlie influences wliich act npon them, 
the laws they obey, and of the external changes which these 
bodies undergo without aflecting their internal constitution. 

Chemistry^ on the contrary, treats of the individual properties 



INTRODUCTION. 13 

of bodies, b}^ whicli, as regards their constitution, they may be 
distinguished one from another ; it also investigates the transfor- 
mations which take place in the interior of a body — transforma- 
tions by which the substance of the body is altered and remodeled; 
and lastly, it detects and classifies the laws by w^hich chemical 
changes are regulated. 

Natural History^ is that branch of physical science which 
treats of organized bodies ; it comprises three divisions, the one 
mechanical — the anatomy and dissection of plants and animals ; 
the second, chemical — animal and vegetable chemistry ; and the 
third, explanatory — ^physiology. 

Astronomy teaches the knowledge of the celestial bodies. It is 
divided into Spherical and Physical astronomy. The former 
treats of the appearances, magnitudes, distances, arrangements, 
and motions of the heavenly bodies ; the latter, of their consti- 
tution and physical condition, their mutual influences and actions 
on each other, and generally, seeks to explain the causes of the 
celestial phenomena. 

Again, one most important use of natural science, is the appli- 
cation of its laws either to technical purposes — m^echanics^ tech- 
nical chemistry^ j9A«777z<xcy, &c. / to the phenomena of the 
heavenly bodies — -physical astronomy j or to the various objects 
which present themselves to our notice at or near the surface of 
the Q2iYt\\—jjhy Steal georjraphy^ meteorology — and we may add 
geology also, a science which has for its object to unfold the 
history of our planet from its formation to the present time. 

Natural philosophy is a science of observation and expervment^ 
for by these two modes we deduce the varied information we 
have acquired about bodies ; by the former we notice any 
changes tliat trans})ire in the condition or relations of any body 
as they spontaneously arise without interference on our i)art ; 
whereas, in the performance of an experiment, we [)urposely 



14 ELEMENTS OF AN-ALTTICAL MECHAI^ICS. 

alter the natural arrangement of things to bring about some par- 
ticular condition that we desire. To accomplish this, we make 
use of appliances called philosophical or chemical ajpjparatus^ the 
proper use and application of which, it is the office of Ex^peri- 
mental Physics to teach. 

If we notice that in winter water becomes converted into ice, 
we are said to make an observation ; if, by means of freezing 
mixtures or evaporation, we cause water to freeze, we are then 
said to perform an experiment. 

These experiments are next subjected to calculation, by which 
are deduced what are sometimes called the laws of nature^ or the 
rules that lihe causes will invariahly produce like results. To 
express these laws with the greatest possible brevity, mathematical 
symbols are used. When it is not practicable to represent them 
with mathematical precision, we must be contented with infer- 
ences and assumptions based on analogies, or with probable 
explanations or hypotheses. 

A hypothesis gains in probability the more nearly it accords 
with the ordinary course of nature, the more numerous the 
exj)eriments on which it is founded, and the more simple the 
explanation it offers of the phenomena for which it is intended to 
account. 

PHYSICS OF PONDERABLE BODIES. 

§ 1. — TliQ p^hysical projjerties of bodies are those external signs 
by which their existence is made evident to our minds; the senses 
constitute the medium through which this knowledge is com- 
municated. 

All our senses, however, are not equally made use of for this 
purpose ; we are generally guided in our decisions by the evidence 
of sight and touch. Still sight alone is frequently incompetent, 
as there are bodies which cannot be perceived by that sense, as, 
for example, all colorless gases ; again, some of the objects of 
sight are not substantial, as, the shadow, the image in a mirror, 



INTRODUCTION. 15 

spectra formed by the refraction of the rays of light, &c. 
Touch, on the contrary, decides indubitably as to the existence 
of any body. 

The properties of bodies may be divided into ^imary or prin- 
cipal^ and secondary or accessory. The former, are such as we 
find common to all bodies, and without which we cannot conceive 
of their existing ; the latter, are not absolutely necessary to our 
conception of a body's existence, but become known to us by 
investigation and experience. 

PKIMAIiY PEOPEETIES. 

§ 2. — The primary properties of all bodies are extension and 
impenetrability. 

Extension is that property in consequence of which every body 
occupies a certain limited space. It is the condition of the 
mathematical idea of a body ; by it, the volwne or size of the 
occupied space, as well as its boundary, or figure.^ is determined. 
The extension of bodies is expressed by three dimensions, length, 
breadth, and thickness. The computations from these data, follow 
geometrical rules. 

Impenetrability is evinced in the fact, that one body cannot 
enter into the space occupied by another, without previously 
thrusting the latter from its place. 

A body then, is whatever occupies space, and possesses exten- 
sion and impenetrability. One might be led to imagine that the 
property of impenetrability belonged only to solids, since we see 
them penetrating both air and water ; but on closer observation 
it will be apparent that this property is common to all bodies of 
whatever nature. If a hollow cylinder into which a piston fits 
accurately, be filled with water, the piston cannot be thrust into 
the water, thus showing it to be impenetrable. Invert a glass 
tumbler in any liquid, the air, unable to escape, will prevent the 
liquid from occupying its place, thus proving the impenetrability 



16 



ELEMENTS OF ANALYTICAL MECHANICS, 



of air. The diving-bell affords a familiar illustration of tliis 
property. 

The difSculty of pouring liquid into a vessel having only one 
small hole, arises from the impenetrability of the air, as the 
liquid can run into the vessel only as the air makes its escape. 
The following experiment will illustrate this fact : 

In one mouth of a two- 
necked bottle insert a funnel 
<z, and in the other a siphon h 
the longer leg of which is im- 
mersed in a glass of water. 
Now let water be poured into 
the funnel a^ and it will be 
seen that in proportion as this 
water descends into the vessel 
F, the air makes its escape 



through the tube 



J, as is 



H J 


i , 


^' 


W 


tX 




m 








m 



proved by the ascent of the 
bubbles in the water of the 
tumbler. 



SECONDARY PEOPEE.TIES. 



The secondary properties of bodies are com^i^essibility^ cxpansi- 
lillty, jporosity, divisihility^ and elasticity. 



% 3. — Compressibility is that property of bodies by virtue of 
which they may be made to occupy a smaller space : and cxpansi- 
hility is that in consequence of which they may be made to fill a 
larger, without in either case altering the quantity of matter they 
contain. 

Both changes are produced in all bodies, as we shall presently 
see, by change of temperature ; many bodies may also be reduced 
in bulk by pressure, percussion, &c. 



INTRODUCTION. 17 

§ 4. — Since all bodies admit of compression and expansion, it- 
follows of necessity, that there must be interstices between their 
minutest particles ; and that property of a body by which its 
constituent elements do not completely fill the space within its 
exterior boundary, but leaves holes or pores between them, is 
caXled porosit I/. The pores of one body are often filled with some 
other body, and the pores of this with a third, as in the case of a 
sponge containing water, and the water, in its turn, containing 
air, and so on till we come to the most subtle of substances, 
ether, which is supposed to pervade all bodies and all space. 

In many cases the pores are visible to the naked eye ; in others 
they are only seen by the aid of the microscope, and when so 
minnte as to elude the power of this instrument, their existence 
may be inferred from experiment. Sponge, cork, wood, bread, 
&c., are bodies whose pores are noticed by the naked eye. The 
human skin appears full of them, when viewed with the magni- 
fying glass ; the porosity of water is shown by the ascent of air 
bubbles when the temperature is raised. 

§ 5. — Tlie divisibility of bodies is that property in consequence 
of which, by various mechanical means, such as beating, pound- 
ing, grinding, &c., we can reduce them to particles homogeneous 
to each other, and to the entire mass ; and these again to smaller, 
and so on. 

By the aid of matnematical processes, the mind may be led to 
admit the infinite divisibility of bodies, though their practical 
division, by mechanical means, is subject to limitation. Many 
examples, however, prove that it may be carried to an incredible 
extent. We arc furnished with numerous instances amono^ nat- 
ural objects, whose existence can only be detected by means of 
the most acute senses, assisted by the most powerful artificial 
aids; the size of such objects can only be calculated approxi- 
mately. 

Mechanical subdivisions for purposes connected with the arts 

are exemplified in the grinding of corn, the pulverizing of sul- 

2 



18 ELEMENTS OF ANALYTICAL MECHANICS. 

piiTir, charcoal, and saltpetre, for the manufacture of gunpowder ; 
and Homeopathy affords a remarkable instance of the extended 
application of this property of bodies. 

Some metals, particularly gold and silver, are susceptible of a 
very great divisibility. In the common gold lace, the silver 
thread of which it is composed is covered with gold so attenuated, 
that the quantity contained in a foot of the thread weighs less 
than g-o'oo of a grain. An inch of such thread will therefore 
contain 72J00 ^^ ^ grain of gold; and if the inch be divided into 
100 equal parts, each of which would be distinctly visible to the 
eye, the quantity of the precious metal in each of such pieces 
would be 720^000 o^ 3- gi'ain. One of these particles examined 
through a microscope of 500 times magnifying power, will appear 
600 times as long, and the gold covering it will be visible, having 
been divided into 3,600,000,000 parts, each of which exhibits all 
the characteristics of this metal, its color, density, &c. 

Dyes are likewise susceptible of an incredible divisibility. 
With 1 grain of blue carmine, 10 lbs. of water may be tinged 
blue. These 10 lbs. of water contain about 617,000 drops. Sup- 
posing now, that 100 particles of carmine are required in each 
drop to produce a uniform tint, it follows that this one grain of 
carmine has been subdivided 62 millions of times. 

According to Biot, the thread by which a spider lets herself 
down is composed of more than 5000 single threads. The single 
threads of the silkworm are also of an extreme fineness. 

Our blood, which appears like a uniform red mass, consists of 
small red globules swimming in a transparent fluid called serum. 
The diameter of one of these globules does not exceed the 4000th 
part of an inch : whence it follows that one drop of blood, such 
as would hang from the point of a needle, contains at least one 
million of these globules. 

But more surprising than all, is the microcosm of organized nature 
in the Infusoria, for more exact acquaintance with which we are 
indebted to the unwearied researches of Ehrenberg. Of these crea- 



INTRODUCTION. 19 

tares, wliicli for the most part we can see only by the aid of the 
microscope, there exist many species so small that millions piled on 
each other would not equal a single grain of sand, and thousands 
might swim at once through the eye of the finest needle. The 
coats-of-mail and shells of these animalcules exist in such prodi- 
gious quantities on our earth that, according to Ehrenberg's inves- 
tigations, pretty extensive strata of rocks, as, for instance, the 
smooth slate near Bilin, in Bohemia, consist almost entirely of 
them. By microscopic measurements 1 cubic line of this slate con- 
tains about 23 millions, and 1 cubic inch about 41,000 millions of 
these animals. As a cubic inch of this slate weighs 220 grains, 
187 millions of these shells must go to a grain, each of which 
would consequently weigh about the ytt millionth part of a grain. 
Conceive further that each of these animalcules, as microscopic 
investigations have proved, has his limbs, entrails, &c., the possi- 
bility vanishes of our forming the most remote conception of the 
dimensions of these organic forms. 

In cases where our finest instruments are unable to render us 
the least aid in estimating the minuteness of bodies, or the 
degree of subdivision attained; in other words, when bodies 
evade the perception of our sight and touch, our olfactory nerves 
frequently detect the presence of matter in the atmosphere, of 
which no chemical analysis could afibrd us the slightest inti- 
mation. 

Thus, for instance, a single grain of musk difiuses in a large 
and airy room a powerful scent that frequently lasts for years ; 
and papers laid near musk will make a voyage to the East Indies 
and back without losing the smell. Imagine now, how many par- 
ticles of musk must radiate from such a body every second, in 
order to render the scent perceptible in aU directions, and you 
will be astonished at their number and minuteness. 

In like manner a single drop of oil of lavender, evaporated in a 
spoon over a spirit-lamp, fills a large room with its fragrance for 
a length of time. 



20 ELEMENTS OF ANALYTICAL MECHANICS. 

§ 6. — Elasticity is the name given to that property of bodies, 
by virtue of which they resume of themselves their figure and 
dimensions, when these have been changed or altered by any 
extraneous cause. Different bodies possess this property in very 
different degrees, and retain it with very unequal tenacity. 

The following are a few out of a large number of highly 
elastic solid bodies ; viz., glass, tempered steel, ivory, whale 
bone, &c. 

Let an ivory ball fall on a marble slab smeared with some col- 
oring matter. The point struck by the ball shows a. round speck 
which will have imprmted itself on the surface of the ivory with 
out its spherical form being at all impaired. 

Fluids under peculiar circumstances exhibit considerable elas- 
ticity; this is particularly the case with melted metals, more 
evidently sometimes than in their solid state. The following 
experiment illustrates this fact with regard to antimony and 
bismuth. 

Place a little antimony and bismuth on a piece of charcoal, so 
that the mass when melted shall be about the size of a pepper- 
corn ; raise it by means of a blowpipe to a white heat, and then 
turn the ball on a sheet of paper so folded as to have a raised 
edge all round. As soon as the liquid metal falls, it divides itself 
into many minute globules, which hop about upon the paper and 
continue visible for some time, as they cool but slowly ; the points 
at which they strike the paper, and their course upon it, will be 
marked by black dots and lines. 

The recoil of cannon-balls is owing to the elasticity of the iron 
and that of the bodies struck by them. 

FORCE. 

§ Y.^ — "Whatever tends to change the actual state of a body, in 
respect to rest or motion, is called a force. If a body, for 
instance, be at rest, the influence which changes or tends to 
change this state to that of motion, is called force. Again, if a 



INTRODUCTION. 21 

body be abeady in motion, any cause which urges it to move 
faster or slower, is caMed force. 

Of the actual nature of forces we are ignorant ; we know of 
their existence only by the effects they produce, and with these 
we become acquainted solely through the medium of the senses. 
Hence, while their operations are going on, they appear to us 
always in connection with some body which, in some way or 
other, affects our senses. 

§ 8. — 'We shall find, though not always upon superficial inspec- 
tion, that the approaching and receding of bodies or of their com- 
ponent parts, when this takes place apparently of their own 
accord, are but the results produced by the various forces that 
come under our notice. In other words, that the universally ope- 
rating forces are those of attraction and of repulsion. 

§ 9. — Experience proves that these universal forces are at work 
in two essentially different modes. They are operating either in 
the interior of a body, amidst the elements which compose it, or 
they extend their influence through a wide range, and act upon 
bodies "in the aggregate ; the former distinguished as Atomical 
or Molecular action^ the latter as the Attraction of gravitation. 

§ 10. — Molecular forces and the force of gravitation, often co- 
exist, and qualify each other's action, giving rise to those attrac- 
tions and repulsions of bodies exhibited at their surfaces when 
brought into sensible contact. This resultant action is called the 
force of cohesion or of dissolution^ according as it tends to unite 
different bodies, or the elements of the same body, more closely, 
or to separate them more widely. 

§ 11. — Inertia is yiat principle by which a body resists all 
change of its condition, in respect to rest or motion. If a body be 
at rest, it will, in the act of yielding its condition of rest, while 
under the action of any force, oppose a resistance ; so also, if a 
body be in motion, and be urged to move faster or slower, it will, 



22 ELEMENTS OF ANALYTICAL MECHANICS. 

during the act of changing, oppose an equal resistance for every 
equal amount of change. We derive our knowledge of this prin- 
ciple solely from experience ; it is found to be common to all 
bodies ; it is in its nature conservative, though passive in charac- 
ter, being only exerted to preserve the state of rest or of particu- 
lar motion which a body has, by resisting all variation therein. 
Whenever any force acts upon a free body, the inertia of the 
latter reacts, and this action and reaction are equal and contrary. 

§ 12. — Molecular action chiefly determines the forms of bodies. 
All bodies are regarded as collections or aggregates of minute ele- 
ments, called atoms^ and are formed by the attractive and repul- 
sive forces acting upon them at immeasurably small distances. 

Several hypotheses have been proposed to explain the constitu- 
tion of a body, and the mode of its formation. The most remark- 
able of these was by Boscovich, about the middle of the last cen- 
tury. Its great fertility in the explanations it affords of the prop- 
erties of what is called tangible matter, and its harmony with the 
laws of motion, entitle it to a much larger space than can be 
found for it in a work like this. Enough may be stated, however, 
to enable the attentive reader to seize its leading features, and to 
appreciate its competency to explain the phenomena of nature. 

1. All matter consists of indivisible and inextended atoms, 

2. These atoms are endowed with attractive and repulsive 
forces, varying both in intensity and direction by a change of dis- 
tance, so that at one distance two atoms attract each other, and at 
another distance they repel. 

3. This law of variation is the same in all atoms. It is, there- 
fore, mutual ; for the distance of atom a from atom J, being the 
same as that of h from a, if a attract u, h must attract a with 
precisely the same force. 

4. At all considerable or sensible distances, these mutual forces 
are attractive and sensibly proportional to the square of the dis- 
tance inversely. It is the attraction called graA)itation. 

5. In the small and insensible distances in which sensible con- 



INTRODUCTION. 23 

tact is observed, and wliich do not exceed the 1000th or 1500th 
part of an inch, there are many alternations of attraction and 
repulsion, according as the distance of the atoms is changed. 
Consequently, there are many situations within this narrow limit, 
in which two atoms neither attract nor repel. 

6. The force which is exerted between two atoms when their 
distance is diminished without end, and is just vanishing, is an 
insuperable repulsion, so that no force whatever can press two 
atoms into mathematical contact. 

Such, according to Boscovich, is the constitution of a material 
atom and the whole of its constitution, and the immediate efficient 
cause of all its properties. 

Two or more atoms may be so situated, in respect to position 
and distance, as to constitute a molecule. Two or more molecules 
may constitute 2i particle. The particles constitute a tody. 

ISTow, if to these centres, or loci of the qualities of what is 
termed matter, we attribute the property called inertia, we have 
all the conditions requisite to explain, or arrange in the order of 
antecedent and consequent, the various operations of the physical 
world. 

Boscovich represents his law of atomical action by what may 
be called an exponential curve. Let the distance of two atoms 





be estimated on the line C A C^ A being the situation of one of 
them, while the other is placed anywhere on this line. AVhen 
placed at ^, for example, we may suppose that it is attracted by 
-4, with a certain intensity. We can re]3resent this intensity by 



24 ELEMENTS OF ANALYTICAL MECHANICS. 

the length of the line i Z, perpendicular to A (7, and can express 
the direction of the force, namely, from i to A^ because it is 
attractive, by placing i I above the axis A C. Should the atom 
be at m, and be repelled by A, we can express the intensity of 
repulsion by m n, and its direction from m towards G by placing 
m n below the axis. 

This may be supposed for every point on the axis, and a curve 
drawn tlirough the extremities of all the perpendicular ordinates. 
This will be the exponential curve or scale of force. 

As there are supposed a great many alternations of attractions 
and repulsions, the curve must consist of many branches lying on 
opposite sides of the axis, and must therefore cross it at 6*, D', 
(7", Z>"^ &c., and at G. All these are supposed to be contained 
within a very small fraction of an inch. 

Beyond this distance, which terminates at G^ the force is 
always attractive, and is called the force of gravitation^ the maxi- 
mum intensity of which occurs at g^ and is expressed by the 
length of the ordinate G'g. Further on, the ordinates are sensibly 
proportional to the square of their distances from ^1, inversely. 
The branch G' G" has the line A (7, therefore, for its asymptote. 

Within the limit A C there is repulsion, which becomes infi- 
nite, when the distance from A is zero ; whence the branch C D* 
has the perpendicular axis, A y, for its asymptote. 

An atom being placed at G^ and then disturbed so as to move 
it in the direction towards A^ will be re23elled, the ordinates of the 
curve being below the axis ; if disturbed so as to move it from 
A^ it will be attracted, the corresponding ordinates being above 
the axis. The point G is therefore a position in which the atom 
is neither attracted nor repelled, and to which it will tend to 
return when slightly removed in either direction, and is called the 
limit of gravitation. 

If the atom be at (7, or C"^ &c., and be moved ever so little 
towards JL, it will be repelled, and when the disturbing cause is 
removed, will fly back ; if moved from A^ it will be attracted 



INTRODUCTION, 



25 




and return. Hence C\ C'\ &c., are positions similar to G^ and are 
called limits of cohesion^ O being termed the last limit of cohe- 
sion. An atom situated at any one of these points will, with that 
at A^ constitute d^jpermanent molecule of the simplest kind. 

On the contrary, if an ato7]% be placed at D\ or D'\ &c., and 
be then slightly disturbed in the direction either from or towards 
A^ the action of the atom at A will cause it to recede still further 
from its first position, till it reaches a limit of cohesion. The 
points D , D'\ &c., are also positions of indiiference, in wdiich the 
atom will be neither attracted nor repel] cd by that at A^ but they 
differ from G^ O 6"', (fee, in this, that an atom being ever so little 
removed from one of them has no disposition to return to it 
again ; these points are called limits of dissolution. An atom 
situated in one of them cannot, therefore, constitute, with that at 
A., a permanent molecule, but the slightest disturbance will de- 
stroy it. 

It is easy to infer, from what has been said, how three, four, 
ifec, atoms may combine to form molecules of different orders of 
complexity, and how these again may be arranged so as by their 
action upon each other to form particles. Our limits will not 
permit us to dwell upon these points, but we cannot dismiss the 
subject without suggesting one of its most interesting conse- 
quences. 

According to the highest authority on the subject, the sun and 
other heavenly bodies have been formed by the gradual subsi- 
dence of a vast nebula, towards its centre. Its molecules forced 



26 ELEMENTS OF ANALYTICAL MECHANICS. 

bj their gravitating action within their neutral limits, are in a 
state of tension, which is the more intense as the accumulation is 
greater ; and the molecular agitations in the sun caused by the 
successive depositions at its surface, make this body, in conse- 
quence of its vast size, the principal and perpetual fountain of 
that incessant stream of ethereal waves which are now generally 
believed to constitute the essence of light and heat. The same 
principle furnishes an explanation of the internal heat of our 
earth which, together with all the heavenly bodies, would doubt- 
less appear self-luminous were the acuteness of our sense of sight 
increased beyond its present limit in the same proportion that the 
sun exceeds the largest of these bodies. The sun far transcends 
all the other bodies of our system in regard to heat and light, and 
is in a state of incandescence simply because of his vastly greater 
size. 

§ 13. — ^The molecular forces are the effective causes which 
hold together the particles of bodies. Through them, the 
molecules approach to a certain distance where they gain a 
position of rest with respect to each other. The power with 
which the particles adhere in these relative positions, is called, 
as we have seen, cohesion. This force is measured by the 
resistance it offers to mechanical separation of the parts of 
bodies from each other. 

The different states of matter result from certain definite 
relations under which the molecular attractions and repulsions 
establish their equilibrium ; there are three cases, viz., two 
extremes and one mean. The first extreme is that in which 
attraction predominates among the atoms ; this produces the 
solid state. In the other extreme repulsion prevails, and the 
gaseous form is the consequence. The mean obtains when 
neither of these forces is in excess, and then matter presents 
itself under the liquid form. 

Let A represent the attraction and H the repulsion, then 



INTRODUCTION. 2T 

the three aggregate forms maj be expressed loj the following 
formulae : 

A> jR solid, 

A <i R gas, 

A = B liquid. 

These three forms or conditions of matter may, for the most 
part, be readily distinguished by certain external peculiarities; 
there are, however, especially between solids and liquids, so 
many imperceptible degrees of approximation, that it is some- 
times difficult to decide where the one form ends and the 
other begins. It is further an ascertained fact that many 
bodies, (perhaps all,) as for instance, water, are capable of 
assuming all three forms of aggregation. 

Thus, supposing that the relative intensity of the molecular 
forces determines these three forms of matter, it follows from 
what has been said above, that this term may vary in the 
same body. 

The peculiar properties belonging to each of these states 
will be explained when solid, liquid, and aeriform bodies come 
severally under our notice. 

§ 14. — ^The molecular forces may so act upon the atoms of 
dissimilar bodies as to cause a new combination or union of 
their atoms. This may also produce a separation between the 
combined atoms or molecules in such manner as to entirely 
change the individual properties of the bodies. Such efforts 
of the molecular forces are called chemical action^ and the 
disposition to exert these efforts, on account of the peculiar 
state of aggregations of the ultimate atoms of different bodies, 
chemical affinity. 

§ 15. — Beyond the last limit of gravitation, atoms attract 
each other : hence, all the atoms of one body attract each 
atom of another, and vice versa: thus giving rise to attrac- 



28 ELEMENTS OF ANALYTICAL MECHANICS. 

tions between bodies of sensible magnitudes through sensible 
distances. The intensities of these attractions are proportional 
to the number of atoms in the attracting body directly, and 
to the square of the distance between the bodies inversely. 

§16. — ^The term universal grawitation is applied to this force 
when it is intended to express the action of the heavenly 
bodies on each other ; and that of terrestrial gravitation or 
simply gravity^ where we wish to express the action of the 
earth upon the bodies forming with itself one whole. The 
force is always of the same kind however, and varies in 
intensity only by reason of a difference in the number of 
atoms and their distances. Its effect is always to generate 
motion when the bodies are free to move. 

Gravity^ then, is a property common to all terrestrial bodies, 
since they constantly exhibit a tendency to approach the 
earth and its centre. In consequence of this tendency, all 
bodies, unless supported, fall to the surface of the earth, and 
if prevented by any other bodies from doing so, they exert a 
pressure on these latter. 

This is one of the most important properties of terrestrial 
bodies, and the cause of many phenomena, of which a fuller 
explanation will be given hereafter. 

§17. — ^The onass of a body is the number of atoms it con- 
tains, as compared with the number contained in a unit of 
volume of some standard substance assumed as unity. The 
unit of volume is usually a cubic foot, and the standard sub- 
stance is distilled water at the temperature of 38°,75 Fahren- 
heit. Hence, the number of atoms contained in a cubic foot 
of distilled water at 38°, 75 Fahrenheit, is the unit of mass. 

The attraction of the earth upon the atoms of bodies at its 
surface, imparts to these bodies, weight ; and if g denote the 



INTRODUCTION. 29 

weight of a unit of mass, J/", the number of units of mass in 
the entire body, and W, its entire weight, then will 

W = M.g (1) 

§18. — Density is a term employed to denote the degree of 
proximity of the atoms of a body. Its measure is the ratio 
arising from dividing the number of atoms the body contains, 
by the number contained in an equal volume of some standard 
substance whose density is assumed as unity. The standard 
substance usually taken, is distilled water at the temperature 
of 38°,Y5 Fahrenheit. Hence, the weights of equal volumes of 
two bodies being proportional to the number of atoms they 
contain, the density of any body, as that of a piece of gold, is 
found by dividing its weight by that of an equal volume of 
distilled water at 38°,Y5 Fahrenheit. 

Denote the density of any body by D, its volume by F, 
and its mass by M, then will 

M= V.D ......... (ly 

which in Equation (1), gives 

W = V.D.g (2) 

§19. — ^That branch of science which treats of the action of 
forces on bodies, is called Mechanics. And for reasons which 
will be explained in the proper place, this subject will be 
treated under the general heads of Mechanics of Solids, and 
Mecha/iiics of Fluids, 



PART I. 



MECHANICS OE SOLIDS 



SPACE, TIME, MOTION, AND FORCE. 

§20. — Space is indefinite extension, without limit, and contains all 
bodies. 

§21. — Time is any limited portion of duration. We may conceive 
of a time which is longer or shorter than a given time. Time has, 
therefore, magnitude, as well as lines, areas, &c. 

To measure a given time, it is only necessary to assume a certain 
interval of time as unity, and to express, by a number, how often 
this unit is contained in the given time. When we give to this 
number the particular name of the unit, as hour, minute, second, &lc., 
we have a complete expression for time. 

The Instruments usually employed in measuring time are clocks, 
chronometers, and common watches, which are too well known to need 
a description in a work like this. 

The smallest division of time indicated by these time-pieces is the 
second, of which there are GO in a minute, 3600 in an hour, and 
86400 in a day ; and chronometers, which are nothing more than a 
species of watch, have been brought to such perfection as not to vary 
in their rate a half a second in 365 days, or 31536000 seconds. 

Thus the number of hours, minutes, or seconds, between any two 
events or instants, may be estimated with as much precision and 



MECHANICS OF SOLIDS. 31 

ease as the number of yards, feet, or inches between the extremities 
of any given distance. 

Time may be represented by 
lines, by laying off upon a 
given right line AB, the equal 

distances from to 1, 1 to 2, " " ^ " ** ^ ^ '^^^ 

2 to 3, &c., each one of these 
equal distances representing the 
unit of time. 

A second is usually taken as the unit of time, and a foot as the 
linear unit. 

§ 22. — A body is in a state of absolute rest when it continues in the 
same place in space. There is perhaps no body absolutely at rest; 
our earth being in motion about the sun, nothing connected with it 
can be at rest. Rest must, therefore, be considered but as a relative 
term. A body is said to be at rest, when it preserves the same 
position in respect to other bodies which we may regard as fixed. 
A body, for example, which continues in the same place in a boat, 
is said to be at rest in relation to the boat, although the boat itself 
may be in motion in relation to the banks of a river on whose sur- 
face it is floating. 

§23. — A body is in motion when it occupies successively different 
positions in space. Motion, like rest, is but relative. A body is in 
motion when it changes its place in reference to those which we 
may regard as fixed. 

Motion is essentially continuous ; that is, a body cannot pass from 
one position to another without passing through a series of interme- 
diate positions ; a point, in motion, therefore describes a continuous 
line. 

When we speak of the path described by a body, we arc to 
understand that of a certain point connected with the body. Thus, 
the path of a ball, is that of its centre. 

§ 24. — The motion of a body is said to be curvilinear or rectilinear, 
according as the path described is a curve or riffht line. Motion is 



32 ELEMENTS OF ANALYTICAL MECHANICS. 

either uniform or varied. A body is said to have uniform motion 
when it passes over equal spaces in equal successive portions of time : 
and it is said to have varied motion when it passes over unequal 
spaces in equal successive portions of time. The motion is said to 
be accelerated when the successive increments of space in equal 
times become greater and greater. It is retarded when these incre 
ments become smaller and smaller. 

§ 25. — Velocity is the rate of a body's motion. Velocity is mea- 
sured by the length of path described uniformly in a unit of time. 

§ 26, — The spaces described in equal successive portions of time 
being equal in uniform motion, it is plain that the length of path 
described in any time will be equal to that described in a unit of time 
repeated as many times as there are units in the time. Let v denote 
the velocity, t the time, and s the length of path described, then will 

s - v.t, (8) 

If the position of the body be referred to any assumed origin 
whose distance from the point where the motion begins, estimated 
in the direction of the path described, be denoted by /S', then will 

s z^ S + v.t (4) 

Equation (3) shows that in uniform motion, the space described 
is alwaijs equal to the product of the time into the velocity ; that the 
spaces described by different bodies moving with different velocities during 
the same time, are to each other as the velocities ; and that ivheji the 
velocities are the same, the spaces are to each other as the times. 

§ 27. — Differentiating Equation (3) or (4), we find 

ds . . 

dt^"' <^) 

that is to say, the velocity is equal to the first differential co-efficient 
of the space regarded as a function of the time. 

Dividing both members of Equation (3) by f, we have 

7=" (6) 



MECHANICS OF SOLIDS. S3 

which shows that, in uniform motion^ the velocity is equal to the whole 
space divided hy the time in which it is described. 

§ 28. — Matter, in its unorganized state, is inanimate or inert. It 
cannot give itself motion, nor can it change of itself the motion 
which it may have received. 
A body at rest will forever 
remain so unless disturbed 

by something extraneous to ^ ~ ~^* 

itself; or if it be in motion 
in any direction, as from a 

to &, it will continue, after arriving at 6, to move towards c in the 
prolongation of ah ; for having arrived at 6, there is no reason why 
it should deviate to one side more than another. Moreover, if the 
body have a certain velocity at 5, it will retain this velocity unaltered, 
since no reason can be assigned why it should be increased rather 
than diminished in the absence of all extraneous causes. 

If a billiard-ball, thrown upon the table, seem to diminish its 
rate of motion till it stops, it is because its naotion is resisted by 
the cloth and the atmosphere. If a body thrown vertically down- 
ward seem to increase its velocity, it is because its weight is inces- 
santly urging it onward. . If the direction of the motion of a stone, 
thro\vn into the air, seem continually to change, it is because the 
weight of the stone urges it incessantly towards the surface of the 
earth.' Experience proves that in proportion as the obstacles to a 
body's motion are removed, will the motion itself remain unchanged. 

When a body is at rest, or moving with uniform motion, its 
inertia is not called into action. 

§ 29. — A force has been defined to be that which changes or tends 
to change the state of a body in respect to rest or motion. Weight 
and Heat are examples. A body laid upon a table, or suspended 
from a fixed point by means of a thread, would move under the 
action of its weight, if the resistance of the table, or that of the 
fixed point, did not continually destroy the effort of the weight. A 
body exposed to any source of heat expands, its particles recede 
from each other, and thus the state of the body is changed. 

3 



34 ELEMENTS OF ANALYTICAL MECHANICS. 

When we push or pull a body, be it free or fixed, we experience 
a sensation denominated pressure^ traction^ or, in general, effort. This 
effSrt is analogous to that which we exert in raising a weight. Forces 
are real pressures. Pressure may be strong or feeble ; it therefore 
has magnitude, and may be expressed in numbers by assuming a 
certain pressure as unity. The unit of pressure will be' taken to be 
that exerted by the weight of -g-J^-g- part of a cubic foot of distilled 
water, at 38°,75, and is called a pound. 

§ 30. — The intensity of a force is its greater or less capacity to 
produce pressure. This intensity may be expressed in pounds, or in 
quantity of motion. Its value in pounds is called its statical mea- 
sure ; in quantity of motion, its dynamical measure. 

§31. — The point of application of a force, is the material point to 
which the force may be regarded as directly applied. 

§32. — The line of direction of a force is the right line which the 
point of application would describe, if it were perfectly free. 

§ 33. — The effect of a force depends upon its intensity, point of 
application, and line of direction, and when these are given the force 
is known. 

§ 34. — Two forces are equal when substituted, one for the other, 
in the same circumstances, they produce the same effect, or when 
directly opposed, they neutralize each other. 

§35. — There can be no action of a force without an equal and 
contrary reaction. This is a law of nature, and our knowledge of it 
comes from experience. If a force act upon a body retained by a 
fixed obstacle, the latter will oppose an equal and contrary resistance. 
If it act upon a free body, the latter will change its state, and in 
the act of doing so, its inertia will oppose an equal and contrary 
resistance. . Action and reaction are ever equal, contrary and simulta- 
neous. 

§36. — If a free body be drawn by a thread, the thread ^s\\\ stretch 
and even break if the action be too violent, and this will the more 
probably happen in proportion as the body is more massive. If a 



MECHANICS OF SOLIDS. 



35 



body be suspended by means of a vertical chain, and a Tveighing 
spring be interposed in the line of traction, the graduated scale of 
the spring will indicate the weight of the 
body when the latter is at rest ; but if 
the upper end of the chain be suddenly 
elevated, the spring will immediately bend 
more in consequence of the resistance 
opposed by the inertia of the body while 
acquiring motion. "When the motion ac- 
quired becomes uniform, the spring will 
resume and preserve the degree of flexure 
which it had at rest. If now, the motion 
be checked by relaxing the effort applied 
to the upper end of the chain, the spring 
will unbend and indicate a pressure less 
than the weight of the body, in conse- 
quence of the inertia acting in opposition to the retardation. The 
oscillations of the spring may therefore serve to indicate the varia- 
tions in the motions of a body, and the energy of its force of 
inertia, which acts against or with a force, according as the velo- 
city is increased or diminished. 




§37. — Forces produce various effects according to circumstances. 
They sometimes leave a body at rest, by balancing one another, 
through its intervention ; sometimes they change its form or break 
it ; sometimes they impress upon it motion, they accelerate or retard 
that which it has, or change its direction ; sometimes these effects are 
produced gradually, sometimes abruptly, but however produced, they 
require some definite time, and are effected by continuous degrees. If 
a body is sometimes seen to change suddenly its state, either in 
respect to the direction or the rate of its motion, it is because the 
force is so great as to produce its effect in a time so short as to 
make its duration imperceptible to our senses, yet some definite por- 
tion of time is necessary for the change. A ball fired from a gun 
will break through a pane of glass, a piece of board, or a sheet of 
paper, when freely suspended, with a rapidity so great as to call into 



36 ELEMENTS OF ANALYTICAL MECHANICS. 

action a force of inertia in the parts which remain, greater than 
the molecular forces which connect the latter with those torn away. 
In such cases the effects are obvious, while the times in which 
they are accomplished are so short as to elude the senses : and yet 
these times have had some definite duration, since the changes, corres- 
ponding to these effects, have passed in succession through their differ- 
ent degrees from the beginning to the endiug. 

§ 38. — Forces which give or tend to give motion to bodies, are 
called motive forces. The agent, by means of which the force is 
exerted, is called a Motor. 

§39. — The statical measure of forces 
may be obtained by an instrument called 
the Dynamometer, which in principle does 
not differ from the spring balance. The 
dynamical measure will be explained fur- 
ther on. 

§ 40. — When a force acts against a point 
in the surface of a body, it exerts a pres- 
sure which crowds together the neighbor- 
ing particles ; the body yields, is compress- 
ed and its surface indented ; the crowded 

particles make an effort, by their molecular forces, to regain their 
primitive places, and thus transmit this crowding action even to the 
remotest particles of the body. If these latter particles are fixed or 
prevented by obstacles from moving, the result will be a compression 
and change of figure throughout the body. If, on the contrary, these 
extreme particles are free, they will advance, and motion will be com- 
municated by degrees to all the parts of the body. This internal motion, 
the result of a series of compressions, proves that a certain time is 
necessary for a force to produce its entire effect, and the error of 
supposing that a finite velocity may be generated instantaneously. 
The same kind of action will take place when the force is employed 
to destroy the motion which a body has already acquired ; it will 
first destroy the motion of the molecules at and nearest the point of 
action, and then, by degrees, that of those which are more remote 
in the order of distance. 




MECHANICS OF SOLIDS. 37 

The molecular springs cannot be compressed without reacting in a 
contrary direction, and with an equal effort. The agent which presses 
a body will experience an equal pressure ; reaction is equal and con- 
trary to action. In pressing the finger against a body, in pulling it 
with a thread, or pushing it with a bar, we are pressed, drawn, or 
pushed in a contrary direction, and with an equal effort. Two weigh- 




ing springs attached to the extremities of a chain or bar, will indicate 
the same degree of tension and in contrary directions when made to 
act upon each other through its intervention. 

In every case, therefore, the action of a force is transmitted through 
a body to the ultimate point of resistance, by a series of equal and 
contrary actions and reactions which neutralize each other, and which 
the molecular springs of all bodies exert at every point of the right 
line, along which the force acts. It is in virtue of this property of 
bodies, that the action of a force may be assumed to be exerted at 
any 2^oint in its line of direction within the houndary of the body. 

§41. — Bodies being more or less extensible and compressible, when 
interposed between the force and resistance, will be stretched or 
compressed to a certain degree, depending upon the energy with which 
these forces act; but as long as the force and resistance remain the 
same, the body having attained its new dimensions, will cease to 
change. On this account, we may, in the investigations which follow, 
assume that the bodies employed to transmit the action of forces from 
one point to another, are inextensible and rigid. 

WORK. 

§42.— To worJc is to overcome a resistance continually recurring 
along some path. Thus, to raise a body through a vertical height, its 
weight must be overcome at every point of the vertical path. If a 



38 ELEMENTS OF ANALYTICAL MECHANICS. 

body fall through a vertical height, its weight overcomes its inertia at 
every point of the descent. To take a shaving from a board with a 
plane, the cohesion of the wood must be overcome at every point 
along the entire length of the path described by the edge of the chisel. 

§43. — The resistance may be constant, or it may be variable. In 
the first case, the quantity of work performed is the constant resistance 
taken as many times as there are points at which it has acted, and 
is measured by the product of the resistance into the path described 
by its point of application, estimated in the direction of the resistance. 
"When the resistance is variable, the quantity of work is obtained by 
estimating the elementary quantities of work and taking their sum. 
By the elementary quantity of work, is meant the intensity of the 
variable resistance taken as many times as there are points in the 
indefinitely- small path over which the resistance may be regarded as 
constant; and is measured by the intensity of the resistance into the 
differential of the path, estimated in the direction of the resistance. 

§ 44. — In general, let P denote any variable resistance, and 5 the 
path described by its point of application, estimated in the direction 
of the resistance; then will the quantity of work, denoted by Q^ be 
given by 

q=jp.d, ....... (7) 

which integrated between certain limits, will give the value of Q. 

g 45. — The simplest kind of work is that performed in raising a 
weight through a vertical height. It is taken as a standard of com- 
parison, and suggests at once an idea of the quantity of work 
expended in any particular case. 

Let the weight be denoted by PT, and the vertical height by R\ 

then will 

Q^- W.H (8). 

If W become one pound, and H one foot, then will | 

and the unit of work is, therefore, the unit of force, one pound, 
exerted over the unit of distance, one foot; and is measured by s, 



MECHANICS OF SOLIDS. 



square of which the adjacent sides are respectively one foot and one 
pound, taken from the same scale of equal parts. 

§ 46. — To illustrate the use of Equation (7), let 
it be required to compute the quantity of work 
necessary to compress the spiral spring of the 
common spring balance to any given degree, say 
from the length AB to DB. Let the resistance 
vary directly as the degree of compression, and 
denote the distance AD' by x ; then will 

F= C.x; 

in which C denotes the resistance of the spring 
when the balance is compressed through the dis- 
tance unity. 

This value of P in Equation (7), gives 




c-^+c, 



q^ fP .dx-fC.xdx 
which integrated between the limits a; = and x = AD = a, gives 



Q = C- 



2' 



Let C=10 pounds, a = ^ feet; then will 

^ = 45 units of work, 

and the quantity of work will be equal to that required to raise 
45 pounds through a vertical height of one foot, or one pound 
through a height of 45 feet, or 9 pounds through 5 feet, or 5 
pounds through 9 feet, &c., all of which amounts to the same thing, 

§47. — A mean resistance is that which, multiplied into the entire 

path described in the direction of the resistance, will give the entire 

quantity of work. Denote this by i?, and the entire path by s, 
and from the definition, we have 

E.s = fP.ds] 



whence, 



P = 



fP.ds 



(9). 



40 



ELEMENTS OF ANALYTICAL MECHANICS. 



That is, the mean resistance is equal to the entire work, divided 
by the entire path. 

In the above example the path being 3 feet, the mean resistance 
would be 15 pounds. 



§48. — Equation (7) shows that the quantity of work is equal to 
the area included between the path 5, in the direction of the 
resistance, the curve whose ordinates are the different values of P, and 
the ordinates which denote the extreme resistances. Whenever, 
therefore, the curve which connects the resistance with the path is 
known, the process for finding the quantity of work is one of 
simple integration. 

Sometimes this law cannot be found, and the intensity of the 
resistance is given only at certain points of the path. In this case 
we proceed as follows, viz. : At the several points of the path 
where the resistance is known, erect ordinates equal to the cor- 
responding resistances, and connect their extremities by a curved 
line ; then divide the path described into any even number of equal 
parts, and erect the ordinates 
at the points of division, and 
at the extremities ; number 
the ordinates in the order 
of the natural numbers; add 
together the extreme ordinates, 
increase this sum by four times 
that of the even ordinates and 
twice that of the uneven ordi- 
nates, and multiply by one-third 
of the distance between any 
two consecutive ordinates. 

Demonstration : To compute the area comprised by a curve, any 
two of its ordinates and the axis of abscisses, by plane geometry, 
divide it into elementary areas, by drawing ordinates, as in the 
last figure, and regard each of the elementary figures, Ci e^ r^ ^i, 
e^ e^ 7*3 ra, &;c., as trapezoids ; it is obvious that the error of 




MECHANICS OF SOLIDS. 



41 






772, n, + en r. 



3 ^1 ^2, 



this supposition will be less, in proportion 
as the number of trapezoids between given 
limits is greater. Take the first two trape- 
zoids of the preceding figure, and divide the 
distance c^^s i into thr^e equal parts, and at 
the points J -of division, erect the ordinates 
m n, Wi rii 5 the area computed from the three 
trapezoids e^ m n r^, m m^ n^ n, m^ e^ r^ n^, will 
be more accurate than if computed from the 

two &j ^2 '2 ' H 2 3 3 2* 

The area by the three trapezoids is 

Ci m X [- mmi + m^ e^ 

But by construction, 

6x711 = mm^ = rriie^ =z ^Cii 

and the above may be written, 

^ ^1 ^2 (^1 7*1 + 2 m n + 2 mi ni + e^ 7-3), 

but in the trapezoid mm^nx 7^, 

2 m 71 + 2 mi 7ii = 4 ^-j 7*2, very nearly ; 

whence the area becomes 

i ^1 ^2 (^1 ^1 + 4 e^ r., + ^3 7-3) ; 

the area of the next two trapezoids in order, of the preceding 
figure, will be 

i ^1 e, (?3 rs + 4 64 r4 + e, r,) ; 

and similar expressions for each succeeding pair of trapezoids. 
Taking the sum of these, and we have the whole area bounded by 
the curve, its extreme ordinates, and the axis of abscisses ; or, 

Q = ie,e,[€,r, + 4e,r^ + ^e,r2 + 4:C,r, + 2^57-5 + 4e,r, + e.r,] . (10) 
whence the rule. 



42 ELEMENTS OF ANALYTICAL MECHANICS. 

§49. — By the processes now explained, it is easy to estimate the 
quantity of work of the weights of bodies, of the resistances due to 
the forces of affinity which hold their elements together, of their 
elasticity, &c. It remains to consider the rules by which the quantity 
of work of inertia may be computed. Inertia is exerted only during 
a change of state in respect to motion or rest, and this brings us 
to the subject of varied motion. 

VARIED MOTION. 

§ 50. — Varied motion has been defined to be that in .which unequal 
spaces are described in equal successive portions of time. In this 
kind of motion the velocity is ever varying. It is measured at any 
given instant by the length of path it would enable a body to 
describe in the first subsequent unit of time, were it to remain 
unchanged. Denote the space described by s, and the time of its 
description by t. 

However variable the motion, the velocity may be regarded as 
constant during the indefinitely small time, dt. In this time the 
body will describe the small space ds ; and as this space is des- 
cribed uniformly, the space described in the unit of time would, 
were the velocity constant, be ds repeated as many times as the 
unit of time contains dt. Hence, denoting the value of the velo- 
city at any instant by v, we have 

v^dsx^ 



or, 

ds_ 
dt 



^ = ^ (11) 



§ 51. — Continual variation in a body's velocity can only be pro- 
duced by the incessant action of some force. The body's inertia 
opposes an equal and contrary reaction. This reaction is directly 
proportional to the mass of the body and to the amount of change 
in its velocity ; it is, therefore, directly proportional to the product 
of the mass into the increment or decrement of the velocity. The 
product of a mass into a velocity, represents a quantity of motion. 



MECHANICS OF SOLIDS. 43 

The intensity of a motive force, at any instant, is assumed to be 
measured by the quantity of motion which this intensity can generate 
in a unit of time. 

The mass remaining the same, the velocities generated in equal 
successive portions of time, by a constant force, must be equal to 
each other. However a force may vary, it may be regarded as 
constant during the indefinitely short interval dt ; in this time it will 
generate a velocity dv^ and were it to remain constant, it would 
generate in a unit of time, a velocity equal to dv repeated as many 
times as dt is contained in this unit; that is, the velocity generated 
would be equal to 

dt dt ' 

and denoting the intensity of the force by P, and the mass by J/, 
we shall have 

P = M.^ (12) 

dt 

Again, differentiating Equation (11), regarding t as the independent 

variable, we get, 

d:^s 
dv =z -— : 
dt ' 

and this, in Equation (12), gives 

P = M,^ (13) 

dt^ 

From Equation (11), we conclude that in varied motion, the velocity/ 
at any instant is equal to the first differential co-efficient of the space 
regarded as a function of the time. 

From Equation (12), that the intensity of any motive force, or of 
the inertia it develops, at any instant, is measured by the 2^^'oduct of 
the niass into the first differential co-effcient of the velocity regarded as 
a function of the time. 

And from Equation (13), that the intensity of the motive force or 
of inertia, is measured by the product of the mass into tJie second 
differential co-efficient of the space regarded as a function of the time. 



44: ELEMENTS OF ANALYTICAL MECHANICS. 

§52. — To illustrate. Let there be the relation 

s = at'^-\-bfi (14) 

required the space described in three seconds, the velocity at the end 
of the third second, and the intensity of the motive force at the same 
instant. 

Differentiating Equation (14) twice, dividing each result by dt^ and 
multiplying the last by Jf, we find 

J = v = 3a^2 4. 2&^ . . . . (15) 

if.— == P = J!f[6a^ + 26] . . . (16) 

Make a = 20 feet, 5 =: 10 feet, and t = 3 seconds, we have, 
from Equations (14), (15), and (16), 

s r= 20 . 33 4- 10 . 32 = 630 feet ; 

v= 3.20.32 + 2.10.3 = 600 feet; 

F = M(Q . 20 . 3 + 2 . 10) =: 380 . i/". 

That is to say, the body will move over the distance 630 feet in 
three seconds, will have a velocity of 600 feet at the end of the 
third second, and the force will have at that instant an intensity 
capable of generating in the mass M, a velocity of 380 feet in one 
second, were it to retain that intensity unchanged. 

§ 53. — Dividing Equations (12) and (13) by M, they give 

(17) 



p 

M~ 


dv 
"dt 


P 


d'^s 
dfl 



(18) 



The first member is the same in both, and it is obviously that 
portion of the force's intensity which is impressed upon the unit of 
mass. The second member in each is the velocity impressed in the 
unit of time, and is called the acceleration due to the motive force. 



MECHANICS OF SOLIDS. 45 

§54. — From Equation (11) we have, 

ds = V ,dt (19) 

multiplying this and Equation (12) together, there will result, 
P .ds = M.v.dv . . . . (20) 



and integrating, 



fP.,ds = ^ ..... (21) 



The first member is the quantity of work of the motive force, 
which is equal to that of inertia; the product M.v^, is called the 
living force of the body whose mass is M. "Whence, we see that 
the work of inertia is equal to half the living force ; and the living 
force of a body is double the quantity of work expended by its inertia 
while it is acquiring its velocity. 

§ 55. — If the force become constant and equal to F, the motion 
will be uniformly varied, and we have, from Equation (18), 

F _d^ 
M ~ If' 

Multiplying by dt and integrating, we get 

|.< = |+C = .+ (7 . . (22) 

and if the body be moved from rest, the velocity will be equal to 
zero when t is zero ; whence C =z 0, and 

^•^ = ^ (23) 

Multiplying Equation (22) by dt, after omitting C from it, and 
integrating again, we find 

F ^ 

and if the body start from the origin of spaces, C'' will be zero, and 
F <2 



M 2 



= * (24) 



46 ELEMENTS OF ANALYTICAL MECHANICS. 

Making t equal to one second, in Equations (24) and (23), and 
dividing the last by the first, we have 

1 - -L 

2 ~ y' 

or, -y = 25 ....... . (25) 

That is to say, the velocity generated in the first unit of time is 
measured hy double the space described in acquiring this velocity. 
Equations (23), (24), and (25) express the laws of constant forces. 

§ 56. — The dynamical measure for the intensity of a force, or the 
pressure it is capable of producing, is assumed to be the effect this 
pressure can produce in a unit of time, this effect being a quantity 
of motion, measured by the product of the mass into the velocity 
generated. This assumed measure must not be confounded with the 
quantity of work of the force while producing this effect. The 
former is the measure of a single pressure; the latter, this pressure 
repeated as many times as there are points in the path over which 
this pressure is exerted. 

Thus, let the body be moved from A to 
j5, under the action of a constant force, in 
one second; the velocity generated will. 
Equation (25), be 2AB. Make BC=2AB, ' 
and complete the square BCFK BE will 
be equal to v ; the intensity of the force 
will be M.v\ and the quantity of work, 
the product of M .v by AB^ or by its 
equal ^ v ; thus making the quantity of 
work "I M v"^^ or the mass into one half the 
square BF\ which agrees with the result obtained from Equation (21). 

EQUILrBEIUM. 

g 57. — Equilibrium is a term employed to express the state of 
two or more forces which balance one another through the interven- 
tion of some body subjected to their simultaneous action. When 
applied to a body, it means that the body is at rest. - 




MECHANICS OF SOLIDS. 47 

We must be careful to distinguish between the extraneous forces 
"which act upon a bodj, and the forces of inertia which they may, or 
may not, develop. 

If a body subjected to the simultaneous action of several extraneous 
forces, be at rest, or have uniform motion, the extraneous forces are 
in equilibrio, and the force of inertia is not developed. If the body 
have varied motion, the extraneous forces are not in equilibrio, but 
develop forces of inertia which, with the extraneous forces, are in 
equilibrio. Forces, therefore, including the force of inertia, are ever 
in equilibrio ; and the indication of the presence or absence of the 
force of inertia, in any case, shows that the body is or is not chang- 
ing its condition in respect to rest or motion. This is but a conse- 
quence of the universal law that every action is accompanied by an 
equal and contrary reaction. 

THE COKD. 

§ 58. — A cord is a collection of material points, so united as 
to form one continuous and flexible line. It will be considered, 
in what immediately follows, as perfectly flexible, inextensihle, and 
without thickness or weight. 

§ 59. — By the tension of a cord is meant, the effort by which any 
two of its adjacent particles are urged to separate from each other. 

§ 60. — Two equal forces, P and P\ applied at the extremities 
A, A^ of a straight cord, and 
acting in opposite directions 

from its middle point, will ^ -j'- i^ ^ 

maintain each other in equi- 
librio. For, all the points 

of the cord being situated on the line of direction of the forces, any 
one of them, as 0, may be taken as the common point of applica- 
tion without altering their effects ; but in this case, the forces being 
equal will, §34, neutralize each other. 



48 



ELEMENTS OT ANALYTICAL MECHANICS. 



§61. — If two equal forces, P and P', solicit in opposite directions 
the extremities of the cord 
A A^, the tension of the cord 

will be measured by the in- T' J[ A P 

tensity of one of the forces. 
For, the cord being in this 

case in equilibrio, if we suppose any one of its points as 0, to become 
fixed, the equilibrium will not be disturbed, while all communica- 
tion between the forces will be intercepted, and either force may 
be destroyed without affecting the other, or the part of the cord on 
which it acts. But if the part ^0 of the cord be attached to a 
fixed point at 0, and drawn by the force P alone, this force must 
measure the tension. 

THE MUFFLE. 




/ii 



§ 62. — Suppose A^ A'^ B, B\ 6zc., to be several small wheels or 
pulleys perfectly free 
to move about their 
centres, which, con- 
ceive for the present 
to be fixed points. 
Let one end of a cord 
be fastened to a fixed 
point C, and be 
wound around the 
pulleys as represent- 
ed in the figure; to the other extremity, attach a weight w. The 
weight w will be maintained in equilibrio by the resistance of the 
fixed point (7, through the medium of the cord. The tension of the 
cord will be the same throughout its entire length, and equal to 
the weight lo ; for, the cord being perfectly flexible, and the wheels 
perfectly free to move about their centres, there is nothing to 
intercept the free transmission of tension from one end to the other. 

Let the points s and r of the cord be supposed for a moment, 
fixed ; the intermediate portion s r may be removed without affecting 



MECHANICS OF SOLIDS. 49 

the tension of the cord, or the equilibrium of the weight w. At 
the point r, apply in the direction from r to a, a force whose inten- 
sity is equal to the tension of the cord, and at s an equal force 
acting in the direction from s io h\ the points r and s may now be 
regarded as free. Do the same at the points s\ r\ s^\ r^\ s'^' and 
r^^^, and the action of the weight w, upon the pulleys A and A^ will 
be replaced by the four forces at 5, s^, s^^ and s^^^, all of equal in- 
tensity and acting in the same direction. 

Now, let the centres of the pulleys A and A'' be firmly con- 
nected with each other, and with some other fixed point as m, in 
the direction of JBA produced, and suppose the pulleys diminished 
indefinitely, or reduced to their centres. Each of the points A and 
A^ will be solicited in the same direction, and along the same line, 
by a force equal to 2w, and therefore the point m, by a force 
equal to 4lW. 

Had there been six pulleys instead of four, the point m would 
have been solicited by a force equal to Qw, and so of a greater 
number. That is to say, the point 771 would have been solicited by 
a force equal to w, repeated as many times as there are pulleys. 

If the extremity C of the cord had been connected with the point 
wi, after passing round a fifth pulley at (7, the point m would 
have been subjected to the action of a force equal to 6w ; if 
seven pulleys had been employed, it would have been urged by a 
force Ifw ; and it is therefore apparent, that the intensity of the 
force which solicits the point m, is found by multiplying the tension 
of the cord^ or weight w, by the number of pulleys. 

This combination of the cord with a number of wheels or pulleys, 
is called a muff,e. 

§ 63. — Conceive the point m to be transferred to the position 
rnf or m'\ on the line AB. The centres of the pulleys A^ A\ &c., 
being invariably connected with the point m, will describe equal 
paths, and each equal to imn\ or mm'\ so that each of the parallel 
portions of the cord will be shortened in the first case, or length- 
ened in the second, by equal quantities; and if e denote the length 
of the path described by w, n the number of parallel portions of 

3 



50 



ELEMENTS OF ANALYTICAL MECHANICS. 



the Cv»rd, which is equal to the number of pulleys, and f, the change 
in length of the portion uw in consequence of the motion of m, 
we shall have, because the entire length of the cord remains the same, 

n.e = ^ (26) 

The first member of this equation we shall refer to as the change 
in length of cord on the pulleys. 

§64. — The action of any force P, upon a material point, may be 
replaced by that of a muffle, by making the tension of its cord equal 
to the intensity of the given force, divided by the number of parallel 
portions of the cord. 

EQUILIBRIUM OF A RIGID SYSTEM. 

§ 65. — Let M represent a collection of material points, united in 
any mamier whatever, forming a solid body, and subjected to the 
action of several forces, P, P^, P^^, P^''^, &c. ; and suppose these 
forces in equilibrio. 

Find the greatest force w, which will divide each of the given 
forces without a remamder ; replace the force P by a muffle, having 




a number of pulleys denoted by — ; the tension of the cord will 

w 



MECHANICS OF SOLIDS. 



51 



be denoted "by w. Do the same for each of the forces, and we 
shall have as many muffles as there are forces, and all the cords 
will have the same tension. 

Let the several cords be united into one, as represented in the 
figure, one end being attached at C, the other acted upon by a weight 
equal to the force w. The action upon the body will remain un- 
changed, that is, the substituted forces, including w, will be in equi- 
librio. 

In this state of the system, let a force Q be applied to put the 
body in motion, and at the instant motion begins, withdraw this 
force and stop the motion before the equilibrium of the forces is des- 
troyed. The points of application of 
the original forces will each have 
described an indefinitely small path, 
as mn. Let mrbe the projection 
of this path upon the original direc- 
tion of the force, and denote the 
length of this projection by e. Join 
the point n with any point o, on 
the direction of the force and at 
some definite distance from m. From the triangle onr, we have 




on 



—2 —2 

or + nr ; 



the displacement being indefinitely small, nr may be neglected in 

— 2 

comparison with or , being an indefinitely small quantity of the second 
order ; hence, 



and. 



on = or. 



om — on = om — or = e. 



But the number of pulleys in the muffle which acts along the 
direction of the force P is. 



hence, the change in the length of the cord on the pulleys of this 



52 ELEMENTS OF ANALYTICAL MECHANICS. 

muffle, caused by the slight motion of the point of application of the 
force P, will, since the centre of the pulley B is fixed, be 

P.e 



and denoting by e\ e'\ e"\ &;c., the projections of the paths described 
by the points to which the forces P^, F', P''\ &c., are respectively 
applied, on the original directions of these forces, we shall have 

P'.e' P'\e'' P''\ef'' , 

■ 5 , • ■ , &C., 

w w • w 

for the corresponding changes in the length of the cord on the other 
muffles. 

In all these changes, the cord being inextensible, its entire length 
remains the same, and if the change in length which the portion uw 
undergoes be denoted by f, we shall have 

— (P. e + P'. e' + P". e" + P'", e'" + &c.) + 1 = . . (27) 
w 

This equation expresses the algebraic sum of all the changes in 
the lengths of the several parts of the cord, between the points of 
application, and the fixed points towards which the points of applica- 
tion are solicited ; the effect of these changes being to shorten some 
and lengthen others, some of the terms of Equation (27) must 
be negative. 

Now it is one of the essential properties of a system of forces 
in equilibrio, that a body subjected to their action is just as free to 
move as though these forces did not exist. The additional force Q^ 
therefore, was wholly employed in overcoming the inertia of the 
body ; it was neither assisted nor opposed by the forces represented 
by the action of the muffles, because these forces balanced each 
other, and the motion was arrested before the points of application 
were sufficiently disturbed to break up the equilibrium. But the 
weight ^^, is one of the forces in equilibrio; and the other forces 
which kept this weight from moving before the application of the 



MECHANICS OF SOLIDS. 53 

force Q, will keep it from moving during the slight disturbance. 

We shall, therefore, have 

1=0, ■ 

and Equation (27) will reduce to, 

Pe 4- F'e' + P"e" + P"'e"' + &c. = ; . . . . (28) 

§ 66. — It may be objected, that the given forces are incommensu- 
rable, and that therefore, a force cannot be found which will divide 
each without a remainder; to which it is answered, that Equation 
(28), being perfectly independent of the value of the weight w^ or 
tension of the cord, this weight may be taken so small as to render 
the remainder after division in any particular case, perfectly inappre- 
ciable. 

§ 67. — The indefinitely small paths m n, m'n\ described by -the 
points of application of the forces, P and P\ during the slight motion 
we have supposed, are called virtual veloci- 
ties ; and they are so called, because, being ^ 
the actual distances passed over by the /] 

points to which the forces are applied, in ^^r\^ 

the same time, they measure the relative ry~^' ^-^' 

rates of motion of these points. The dis- 
tances 7- m and r'm', represented by e and 

e', are therefore, the projections of the virtual velocities upon the 
directions of the forces. These projections may fall on the side 
towards which the forces tend to urge these points, or the reverse, 
depending upon the direction of the motion imparted to the system. 
In the first case, the projections are regarded as positive, and in the 
second, as negative. Thus, in the case taken for illustration, m r is 
positive, and m'r' negative. The products Pe and P'e\ are called 
virtual moments. They are the elementary quantities of work of the 
forces P and P\ The forces are always regarded as positive ; the 
sign of a virtual moment will therefore depend upon that of the 
projection of the virtual velocity. 

§ 68. — Referring to Equation (28), we conclude, therefore, that wheii- 
ever several forces are in equilihrio, the algebraic sum of their virtual 



64: ELEMEl^TS OF ANALYTICAL MECHANICS. 

moments is equal to zero ; and in this consists what is called the prin- 
ciple of virtual velocities. 

§69. — Conversely, if in any system of forces, the algebraic sum 
of the virtual moments be equal to zero, the forces will be in equi- 
librio. For, if they be not in equilibrio, some, if not all the points 
of application will have a .motion. Let q, q\ q"^ &c., be the pro- 
jections of the paths which these points describe in the first instant 
of time, and ^, Q\ Q'\ &c., the intensities of such forces as will, 
when applied to these points in a direction opposite to the actual 
motions, produce an equilibrium. Then, by the principle of virtual 
velocities, we shall have 

Pe + P'e' + P"e'^ + &c. -{- Qq + Q'q' + Q"q'' + &c. = 

But by hypothesis, 

Pe + P'e' + P"e" + &c. = 0, 
and hence, 

Qq + Q'q' + Q"q" + &c. = . . , (28)' 

Now, the forces t^, Q\ Q'\ &c., have each been applied in a direc- 
tion contrary to the actual motion ; hence, all the virtual moments in 
Equation (28)' will have the negative sign ; each term must, therefore, 
be equal to zero, which can only be the case by making Q^ Q', (^\ 
&c., separately equal to zero, since by supposition the quantities 
denoted by q^ q\ q'\ are not so. We therefore conclude, that when 
the algebraic sum of the virtual moments of a system of forces is 
equal to zero, the forces will be in equilibrio. 

Whatever be its nature, the effect of a force will be the same if 
we attribute its effort to attraction between its point of application 
and some remote point assumed arbitrarily and as fixed upon its line 
of direction, the intensity of the attraction being equal to that of the 
force. Denote the distance from the po'nt of application of P, to 
that towards which it is attracted by jt?, and the corresponding dis- 
tances in the case of the forces P', P'\ &c., by J9', jp'\ &;c., respect- 
ively ; also, let ^p^ 5p\ Sp'\ &c., represent the augmentation or dimi- 
nution of these distances caused by the displacement, supposed indefi- 
nitely small, then § 65, will 

e — Sp^ e' = 6p\ e" = dp^% &c., 



MECHANICS OF SOLIDS. 56 

and Equation (28) may be written 

PSp + P'5p' + P"6p" + &c. = . . . (29) 

in which the Greek letter 8 simply denotes change in the value of 
the letter written immediately after it, this change arising from the 
small displacement. 

§70. — If the extraneous forces applied to a body be not in equi- 
librio, they will communicate motion to it, and will develop forces of 
inertia in its various elementary masses with which they will be in 
equilibrio ; and if extraneous forces equal in all respects to these forces 
of inertia were introduced into the system, the algebraic sum of the 
virtual moments would be equal to zero. 

But if m denote the mass of any element of the body, s the 
path it describes, its force of inertia will, Eq. (13), be 

m.. ; 

and denoting the projection of its virtual velocity on s by ^5, its vir- 
tual moment will be 

and because the forces of inertia act in opposition to the extraneous 
forces, their virtual moments must have signs contrary to those of 
the latter, and Equation (29) may be written 

2P. op - 277i . ^\ ^5 = ; . . . . (30), 

in which 2 denotes the algebraic sum of the terms similar to that 
written immediately after it. 

PRINCIPLE OF D'ALEMBERT. 

§71. — This simple equation involves the whole doctrine of Mechanics. 
The extraneous forces P, P\ P'\ &c., are called impressed forces. 
The forces of inertia which they develop may or may not be equal to 
them, depending upon the manner of their application. If the impressed 
forces be in equilibrio, for instance, they will develop no force of inertia ; 



66 ELEMENTS OF ANALYTICAL MECHANICS. 

but in all cases, the forces of inertia actually developed will be equal 
and contrary to so much of the impressed forces as determines the 
change of motion. The portions of the impressed forces which deter- 
mine a change of motion are called effective forces ; and from Equation 
(30), we infer that the impressed and effective forces are always in equi- 
librio when the directions of the latter are reversed, and will pre- 
vent all change of motion. This is usually known as D'' Alemherf s 
Frinciple^ and is nothing more than a plain consequence of the law 
that action and reaction are ever equal and contrary. 

This same principle is also enunciated in another way. Since the 
effective forces reversed would maintain the impressed forces in equi- 
librio, and prevent them from producing a change of motion, it 
follows that whatever forces may he lost and gained mnst he in equili- 
brio ; else a motion different from that which actually takes place 
must occur, a supposition which it were absurd to make. 

§ 72. — Equation (30), is of a form too general for easy discussion. 
To transform it, refer the directions of the forces and their points 
of application to three rectangular axes. 

Denote by a, (3, y , the angles which the direction of the force 
P makes with the axes x, y, z, respectively ; by a, 6, c, the angles 
which its virtual velocity makes with the same axes ; and by cp, the 
angle which the virtual velocity and direction of the force make with 
each other, then will 

cos (p =: cos a . cos a + cos h . cos /3 + cos c . cos y. 

Denote by k, the virtual velocity, and multiply the above equation 
by Pk, and we have 

Pk cos <p = Pk cos a . cos a -f- Pk cos h . cos ^ + Pk cos c . cos y ; 

But denoting the co-ordinates of the point of application of P by 
X, 7/, z, we have 

k cos 9 = ^j9 ; k cos a = Sx ] k cos b =z Sy ^ k cos c z=z Sz ; 
and these values substituted above, give 

P .Sp =z P cos a.§x -\- P cos (3 .Sy -f P cos y . Sz. . . (31). 
Similar values may be found for the virtual moments of other forces. 



MECHANICS OF SOLIDS. 5T 

§ 73.— Again 

differentiating and multiplying by m • j^^-j^ we have 

and denoting by 5x, ty, 5z^ the projections of ^5 on x, y, 0, respectively, 
we have 

-^'8s = 5x', -^-ds = Sy ', -^'Ss = 5z, 
ds ds ds 

whence, 

d'^S . C?2^ C?2y C?22 

and similar expressions may be found for the virtual moments of the 
forces of inertia of the other elementary masses. 

§74. — If the, intensity of the force P, be represented by a portion 
of its line of direction, which is the practice in all geometrical 
illustrations of Mechanics, the factors P cos a, P cos /3, and P cos 7, 
in Equation (31), would represent the intensities of forces equal to 
the projections of the intensity P, on the axes ; and regarding these 
as acting in the directions of the axes, the factors (5".r, dy^ and 5z, will 
represent the projections of their virtual velocities, which virtual veloci- 
ties will coincide with that of the force P. 
Again, Equation (32), 

d^x dhi 'dH 

"^•-w '"•*?■ '»-^' 

are forces of inertia in the directions of the axes, and hx, 8y, Sz, are 
the projections of their virtual velocities ; these virtual velocities coincide 
with that of the inertia of m. 

The values of these virtual velocities depend upon the nature of 
the motion. 



58 



ELEMENTS OF ANALYTICAL MECHANICS. 



FREE MOTION. 



§75. — A body is said to have free motion^ when it pursues the 
path and takes the velocity due to the directions and intensities of 
the extraneous and active forces impressed upon it. This motion 
is to be distinguished from that in which the body is constrained, 
by the interposition of some rigid surface or line, to take a path 
different from that which it would describe but for such interposi- 
tion. A body simply falling under the action of its own weight is 
a case of free motion. The same body rolling down an inclined 
surface, is not. 

The most general motion we can attribute to a body is one 
of translation and of rotation combined. A motion of transla- 
tion carries a body from place to place through space, and its 
position, at any instant, is determined by that of some one of its 
elements. A motion of rotation carries the elements of a body 
around some assumed 
point. In this investi- 
gation, let this point 
be that which deter- 
mines the body's place. 

Denote its co-ordi- 
nates by Xj y, Zj and 
those of the element 
w, referred to this point 
as an orighi by a;', 3/', 
z' ; there will thus be 
two sets of axes, and 
supposing them parallel, 
we have 




and diiferentiating, 



X = x^ + x\ 

y ^Vt + y\ 

dx = dXj + dx'^ 
dy = dy, + dy\ 
dz ■=. dz, -f- dz\ 



(33), 



(34). 



MECHANICS OF SOLIDS. 



59 



Demit from m, the per- 
pendiculars mX', m Y\ mZ'^ 
upon the movable axes. 
Denote the first by r', the 
second by r'\ and the third 
by r'". Let 0', 0" , 0"\ 
be the projections of m, on 
the planes xy^ xz^ y z, res- 
pectively. Join the several 
points by right lines as 
indicated in the figure. 

Denote the angle 




Then will 
the triangle m Z' 0" give j 



mZ' 0" by 9, 
mX' 0' by ^, 
m Y' 0'" by 4.. 



the triangle m Y' 0"\ 



the triangle m X' 0', 



1: 



= r sm 9, ) 

=::^'"t'[ (3«). 

— r" cos v]^, ) 

^ = *■' ~^ ''' I (37). 

z' = r' sin -sr, j 



We here have two values of x\ one dependent upon 9, and the 
other upon 4^. If the body be turned through an indefinitely small 
angle about the axis z\ the corresponding increment of x' is obtained 
by differentiating the first of Equations (35) ; and we have 

dx' =^ —r'" sin 9.^9; 
if it be turned through a like angle about the axis y\ the cor- 
responding increment of x' is found by differentiating the first of 

Equations (3G), and 

dx' =z r" cos ^ . d ■\>. 

If these motions take place simultaneously about both axes, the 
above become partial differentials of x\ and we have fur its total 

differential, 

dx' ^^ r" cos ^ .d\ — r'" sin 9 . c/ 9, 



60 



ELEMENTS OF ANALYTICAL MECHANICS. 



replacing r" cos %}/ and r'" sin 9, by their values in the above Equa- 
tions, and we get 

d x' =z z' . d -^ — y' .d(p; 
and in the same way, 

dy' =z x' . d(p — z\ d zi, 
dz' =. if .d-ui — x' .d \^ 

which substituted in Equations (34), give 



(38) 



dx =■ dXi -\- z' .d \ — y' . c?(p, "" 
dy ^^ dy^ + ^' -d 9 — z' . dvS^ 
dz :=. d Zj -\- y' .d 'ui — x'.d-]^. 



(39) 



and because the displacement is indefinitely small, we may write 

. . .(39)' 



6x =z S Xj + z\§-^ — y' .S(p, 
dy =z S y^ + x\ 8(p — z' .Szi, 
Sz — S Zj -\- y' . S r^i — x\ § -^ ; 



and tnese in Equations (31) and (32), give 

P cos a . 5x^ -\- P cos ^ .hj^ ArP cos y . Sz^ 
+ P . (x' ■ cos (3 — y' . cos a) . 09 
+ P . (z' . cos a — x' . cos y) . S-^ 
+ P . (y' . cos 7 — z\ cos ^) . o-si. 



P,5p 



d?s 
df^ 



ci^x 



d^y 



dH 






+ m 
+ m • 



dt'' 




z' . d^x - x' 


.d'^z 


dfi 




y' . d^z - z' 


.d-^y 



df^ 






dh' 



Similar values may be found for P' . 6 p' and m' . ^^-^ • ^s', &c. In 

these values Sx^^ 5y^ and ^0^, will be the same, as also ^9, ^4', and 
^trf, for the first relate to the movable origin, and the latter to the 
angular rotation which, since the body is a solid, must be of equal 



MECHANICS OF SOLIDS. 



61 



values for all the elements ; so that to find the values of the virtual 
moments of the other forces, it will be only necessary suitably to 
accent P, a, /3, 7, x, y, 0, x\ y\ z'. 

These values being found and substituted in Equation (30), we 
shall find, 

2 P. cos a — 2m. -—\ 6x, 

+ ('s P . cos /3 - 2 m . ^^-) 6y, 

+ (2P.cos7-2m~)^0, 
+ r2P. (a;'.Cos/3— /.cosa)— 2m- ^-^ — ^ ~ ^ ' — -\ ^(p 

+ 2P. (z'.cosa— a;'. COS7)— 2 m — ^4. 

+ [2P.y.cos7-g^cos/3)-2m. ^'-^'^~/'-'^'^ ] 5 -us 



). =0.(40) 



But the displacement being entirely arbitrary, the least considera- 
tion will show that Sx^^ Sy^, 6z^, 8(jp, S-^, and S-a, are wholly inde- 
pendent of each other, and this being the case, the principle of inde- 
terminate co-efficients requires that 



d'^x 
2 P . cos a — 2 m • — — - =: 0, 

2 P . cos /3 - 2 m . -^ = 0, 



2 P. cos 7 — 2m. -— - = : 

' di^ ' 



(^) 



2 p. (a;' . cos /3 — y' . cos a) — 2m. -^ — "^^ — ^' 

X, n / / r \ ^ z'.d^X — x'.d'^Z 

2 P . (z . COS a — X. cos 7) —2m- ~ =: 0, 



2 P . (y' . cos 7 — s' . cos /3) — 2 



dfi 

y'. d"z — z'. dhj 
~df- 



= 0. 



- . {.B) 



62 ELEMENTS OF ANALYTICAL MECHANICS. 

§76. — These six equations express either all the circumstances of 
motion attending the action of forces, or all the circumstances of 
equilibrium of the forces, according as inertia is or is not brought 
into action; and the study of the principles of Mechanics is little 
else than an attentive consideration of the conclusions which follow 
from their discussion. 

Equations {A) relate to a motion of translation, and Equations 
(B) to a motion of rotation. They are perfectly symmetrical and 
may be memorized with great ease. 

COMPOSITION AOT) RESOLUTION OF FORCES. 

§77. — When a body is subjected to the simultaneous action of 
several extraneous forces which are not in equilibrio, it will be put 
in motion ; and if this motion may be produced by the action of 
a single force, this force is called the resultatit, and the several forces 
are termed components. 

The resultant of several forces is a single force which^ acting alone^ 
will produce the same effect as the several forces acting simultaneously/ ; 
and the components of a single force, are several forces whose simulta- 
neous action produces the same effect as the single force. 

If, then, several extraneous forces applied to a body, be not in 
equilibrio, but have a resultant, a single force, equal in intensity to 
this resultant, and applied so as to be immediately opposed to it, 
will produce an equilibrium, or what amounts to the same thing, 
if in any system of extraneous forces in equilibrio, the resultant of all 
the forces but one be found, this resultant will be equal in intensity 
and immediately opposed to the remaining force ; otherwise the sys- 
tem could not be in equilibrio. 

Conceive a system of extraneous forces, not in equilibrio, and 
applied to a solid body, and suppose that the equilibrium may be 
produced by the introduction of an additional extraneous force. 
Denote the intensity of this force by i2, the angles which its direc- 
tion makes with the axes x, y and ^, by a, h and c, respectively, 
and the co-ordinates of its point of application by x^ y, z. Tlien, 
because the inertia cannot act, d'^-x^ dhj^ d'^z will be zero, and taking 



MECHANICS OF SOLIDS. 63 

the two origins to coincide, Equations (A) and [B), will give 

Mcosa -{- P' cos a' + P" cos a." + P'" cos a.'" + &c. = 0, 
i2 cos 6 + i"' cos /3' + P" cos /3" + P'" cos /3"' + &c. = 0, 
i2 cos c + P' cos y' + P" cos 7" + P'" cos 7''' + &c. = ; 



B {x co^b — y cos a) + P' {x' cos /3' — y' cos a') 
+ P" {x" cos /3" - y" cos a") + &c. 

R {z cos a — X cos c) + P' (2' cos a' — x' cos 7') 
+ P" {z" COS a" - a;" cos y") + &c. 

jR (y cos c — s cos 6) + P' (y' cos y' — 2' cos (^') 
+ P" (y" cos y" - z" cos /3") + &c. 



1=0, 



Now R is equal in intensity to the resultant of all the other 
forces of the system, or in other words, to the resultant of all the 
original forces ; and if we give it a direction directly opposite to 
that in which it is supposed to act in the above equations, it be- 
comes in all respects the same as that resultant, being equal to it 
in intensity and having the same point of application and line of 
direction. Adding, therefore, 180° to each of the angles a, 6, and c, 
the first terms of the foregoing equations become negative, and 
transposing the other terms to the second member and changing all 
the signs, we have, 



R cos a — P' cos a' + P" cos a" + P'" cos a'" + &c. = X; ^ 
P cos 6 = P' cos /3' + P" cos ^" + P'" cos ^"' + &c. = Y\ 
P cos c = P' cos 7' -f P" cos y" + P'" cos y'" + &c. r= Z. 



(41) 



U (x cos h —y cos a] 



P' {x' cos ^' — y' cos a') 
4-P"(:r"cos/3"-y"cosct") \=L 
4- &c. 



r P' [z' cos a' — X* cos 7') 
P (z cos a — a; cos c) = ^ -f P" (2'' cos a" — x" cos 7") 
+ &c. 



P (y cos c — z cos 6) 



' P' (y' cos 7' - 2' cos iS') ^ 
+P"(y"cos7'^- 2"cos/3") 
4- &c. 



= if; 



=iy. 



(42) 



64: 



ELEMENTS OF ANALYTICAL MECHANICS, 



or, 



i? cos a z=z X. 



R cos h 
B cos c 




R (x cos h — y cos a) = Z, 
R (z cos a — X cos c) =z M^ 
R (y cos c — z cos b) = N, 



(48) 



(44) 



Eliminating R cos a, i2 cos h and ^ cos c, from Equations (44), 
by means of Equations (43), we get, by transposing all the terms to 
the first member 



Xy - Yx + L =0,'^ 
Zx - Xz -I- tV/ = 0, 
Yz - Zy + iV= 0. 



(45) 



These are the equations of a right line. But x, y and z are the 
co-ordinates of the point of application of the resultant; they are, 
therefore, the equations of the line of direction of the resultant R, 
and hence the point of application of the resultant may be taken 
anywhere on this line without changing its effect. Any condition, 
therefore, expressive of the simultaneous existence of these equations, 
will also express the existence of this single line, and of a single 
resultant to the system of foixes. 

§ 78. — To find this condition, multiply the first of these Equations 
by Z, the second by F, the third by JT, and add the products; 
we obtain. 



ZL + YM + XN =z 



(46). 



§79. — Having ascertained, by the verification of this Equation, 
that the forces have a single resultant, its intensity, direction, and 
the equations of its direction may be readily found from Equations 
(43) and (44). 

Squaring each of the group (43), and adding, we obtain. 



i22 (cos2 a + C0S2 b + C0S2 C) = X2 4- r2 + Z\ 



MECHANICS OF SOLIDS. 65 

Extracting the square root and reducing by the relation, 
cos2 Qj _j_ cos^ b + cos2 c = 1, 
there will result, 

R = sj X-' + r^ + Z2 (47) 

■which gives the intensity of the resultant, since X, T and Z are 
known. 

Again, from the same Equations, 



X 

cos a = — , 
Jti 



cos h 



cos c = 



Y 
R' 
Z 
B 



(48) 



which make known the direction of the resultant. 
The group of Equations (44) give, 

Xy - Yx -\- :^ P' {cos (3' x' - cos a' y') =0, ] 
Zx-Xz + :2F' (cos a' z' — cos/' a;') = 0, 
F2-Z2/ H-2P'(cos7'/-cos/3'2') = 0. J 

which are the equations of the direction of the resultant. 

PAKAXLELOGKAM OF FOECES. 



(49) 



§ 80. — If all the forces be applied to the same point, this point 
may be taken as the origin of co-ordinates, in wliich case, 

x' =z x" = x'" &c. = 0, 
y' = y" = y'" &c. = 0, 
z' = z" = z'" &c. = 0, 

and the last term in each of Equations (49), will reduce to zero. 
Hence, to determine the intensity, direction and equations of the 



66 



ELEMENTS OF ANALYTICAL MECHANICS. 



line of direction of the resultant, we have. Equations (47), (48) 
and (49), 

B 



X 



cos a = 



cos h = 



B 
Y 

b' 



(50) 



(51) 



cos C = — J 
Jti 



Xy - Yx = 0, 

Zx - Xz =0,[ (52) 

Yz - Zy = 0.} 

The last three equations show that the direction of the resultant 
passes through the common point of application of all the forces, 
which might have been anticipated. 

§81. — Let the forces be now reduced to two, and take the plane 
of these forces as that of xy ', then will 

y' = y" = y'" = &c. = 90° ; = 0, 

the last Equation of group (43) reduces to, 

Z = 0- 



and the above Equations become, 
B 



VX2 + r2 

X 



cos a = 



cos b 



B' 



(53) 
(54) 



B 

cos c = 0, 

Xy - Yx = (55) 

The last is an equation of a right line passing through the 
origin. The direction of ike restiltant will^ therefore, pass through the 
point of application of the forces. The cos c being zero, c is 90°, 
and the direction of the resultant is therefore in the plane of the forces. 



MECHANICS OF SOLIDS. 



67 



Substituting in Equation (53), for X and Y, their values- from 
Equations (41), we obtain, 

R = V {F' cos a' + F" cos a")2 -f (P' cos /3' + P'' cos (3"Y ; 
and since 



C0S2 cc' -f COS^ /3' =: 1, 
cos2 a" + cos2 /3'' = 1, 

this reduces to 



E = V?'2 + P"2 4- 2 P' P" (cos a' cos a" + cos /3' cos (3") j 

denoting the angle made by the directions of the forces by 5, we 

have,* 

cos a,' cos a" + cos /3' cos /3^' = cos 5 ; 




and therefore, 

E = y'p'2 4. p//2 ^ 2P'F" cosS 



(56) 



from which we conclude that the intensity of the resultant is equal 
to that diagonal of a 'parallelogram whose adjacent sides represent the 
directions and intensities of the components^ which passes through the 
point of application. 

§ 82. ^Substituting in Equations (54), the values of X and Z, from 
Equations (41), we have, 

P cos a =: P' cos a' + P" cos ol'\ 
Rcosb = P' cos /3' + P" cos ^'\ 
and because 

cl' = 90° - iS', 

a" = 90° — /3", 

a = 90° - 5, 

these Equations reduce to, 

P cos a = P' cos a' + P" cos a", 
P sin a z= P' sin a' + P" sin a" ; 



68 ELEMENTS OF ANALYTICAL MECHANICS. 

by transposing and squaring, we obtain, 

P"2 cos2 a" = i^2 cos2 a — 2 B F' COS a cos a' + P'2 cos2 a', 
P"2 sin2 a" = 7^2 sin2 a ~ 2 B F' sin a sin a' + P'2 ^^2 „j .^ 

adding and reducing, 

P''2 ^ j^2 _{_ p/2 _ 2EF' cos (a - a') ; 
but, 

a — a' =z the angle RmP' = 9' ; 

hence, by transposition and reduction, 



or, 

1— coscp'r=2sin2l- 
whence, making 



^2 _^ pr2 _ p//2 

^^^^ = 2FF' ' 



P"2_(E_prY ^ {P" J[. B - F')(F' ^- F' -R) ^ 
2BF' ~ 2RF' ' 



B-^ F' + F" „ 

7, - ^^ 



we obtain, 



. , , /{S - F') {S - B) 
smi9'=:y^^ ^^^ (57) 



RF' 



from wliich we see that the direction of the resultant coincides \vith 
the diagonal of the parallelogram described on the lines represent- 
ing the intensities and directions of the forces. 

Thus^ the resultant of any two forces, applied to the same material 
pointy is represented, in intensity and direction, hy that diagonal of a 
parallelogram, constructed upon the sides representing the intensities 
and directions of the two components, which passes through the point 
of application. 

§83. — In the triangle RmP', since F' R is equal and parallel to 
the line which represents the force F", the angle mF'R = 9, is the 
supplement of the angle S, made by the directions of the components, 
and there will result the following equation, 

cos 1 (J = sm -i9 = V ^^ j7-^r. — ', ' ' (58) 



MECHANICS OF SOLIDS. 



m 



Equation (57), will make known the angle made by the direction 
of the resultant with that of either of two oblique components, pro 
vided, the intensities of the components and resultant be known. 

§84. — Also, from the two triangles RmF' and RmJB'\ we find, 
. . F' . sin (J 



sm (p 



sm9' 



R 

F . sin 6 
R 



(59), 




m 



from which the angles made 
by the direction of the result- 
ant with its two components may be found. 

§85. — Let there now be the three forces P, P', P", applied to 
the material point m, in 
the directions m P, m F', 
mP", not in the same plane ; 
the resultant will be repre- 
sented in intensity and direc- 
tion by the diagonal of a 
parallelopipedon, constructed 
upon the lines representing 
the directions and intensities 
of these components. For, 
lay off tlie distances mA, 
m C\ and m E^ proportional 
to the intensities of the com- 
ponents which act in the direction of these lines, and construct the 
parallelopipedon E B ; the resultant of the components P' and P 
will, § 82, be represented by the diagonal m i?, of the parallelogram 
mABC\ and the resultant of this resultant and the remaining com- 
ponent P", will be represented by the diagonal m D oi the parallelo- 
gram EmBD^ which is that of the parallelopipedon. 

§ 8G. — If the forces act at right angles to each other, the parallel- 
opipedon will become rectangular, and the intensity of the resultant, 
denoted by P, will become known from the formula 



\ 


-__ 


— -t 


^ 


Ti^^ 


^.-^ 




— ■- ■ 


D 

>2 


X 


-^~^-~ 


C7\ 





70 ELEMENTS OF ANALYTICAL MECHANICS. 



and if the angles which the 
direction of the resultant 
makes with those of the 
forces P, F' and F", be 
represented by a, b, and c, 
respectively, then will 

F cos a = F, 
Fcosb = F\ 
Bcosc = F". 




Let three lines be drawn through the point of application m', of 
the force F\ parallel to any three rectangular axes x^y^z; and denote 
by a', /3', 7', the angles which 
the direction of this force 
makes with these axes res- 
pectively ; then will 

F' cos a', 
F' cos iS', 
F' cos Yi 




be the components of the force P', in the direction of the axes, and 
they will act along the lines drawn through the point m'. These are 
the same as the terms composing in pr;rt Equations (A), and as the 
effect of the components is identical with that of the resultant, these 
components may always be substituted for the force F\ The same 



df- df- 



dh 



for the forces of inertia, and 

components of this force in the directions of the axes. 



and m-— -5 denote the. 
dt^ 



MECHANICS OF SOLI'DS. 



n 



§ 87. — Uxamples. — 1. Let the point m, be solicited by two forces 
whose intensities are 9 and 5, and whose directions 
make an angle with each other of 57° 30'. Re- 
quired the intensity of the force by which the 
point is urged, and the direction in which it is 
compelled to move. 

First, the intensity ; make in Equation (56), 

P' =: 9, 
F" = 5, 

S = 57° 30' ; 

and there will result, 

B = V 81 + 25 + 90 X 0, 537 = 12,422. 

Again, substituting the values of S, P' P" and R in the first of 
Equations (59), we have, 

. , 5 X sin 57° 30' 




sm 9' = 



12,422 



or. 



)' = 19° 50' 35'' nearly, 



which is the angle made by the direction of the force 9 with that of 
the resultant. 

2. — Eec[uired the angle under which two equal components should 
act, in order that their resultant shall be the n'^ part of either of them 
separately. 

By condition, we have . 



hence, 



P' = P" = uR 



P' J^ p" ^ R _ e _ nR-^ nR + R _ {2n + 1) R 
2 '-^~ 2 - ^ ' 



and, Equation (58), 



sm 



~ \ pi p'f . ' 



72 ELEMENTS OF ANALYTICAL MECHANICS. 

which reduces to 

If n be equal to unity, or the resultant be equal to either force, 

9 = 60°, 
and, § 83, the angle of the components should be 120°. 

3. — Eequired to resolve the force 18 = a, into two components 
whose difference shall be 5 = 6, and whose directions make with 
each other an angle of 38° = S, Also, to find the angle which the 
direction of each component makes with that of the resultant. 

Writing a for i? in Equation (56), we have, 

p/2 _|_ p/'2 _!_ 2P'P" cos S = a2, 

and by condition. 

F' - F" = b . . . . . . (c). 

Squaring the second and subtracting it from the first, we get 

2F'F" {I + cos^) = a2 _ 52. 

which, replacing (1 + cos ^) by 2 cos^ ^ S, reduces to 

a2 — 62 

C0S2 ^S 

This added to the square of the Equation ( c ), gives 



V cos2 ^ 6 

from which and Equation (c) we finally obtain. 



l{^./^- 


- 62 (I — cos2 \S) 


- ^K-y 


COs2|-5 


r(^.n^- 


- 62 (1 - cos2|-(^) 



049, 



/ / (,2 _ J2 1 _ C0S2 h^\ \ 

^" = i (^ V ^T— ^ -V= ^'^^^' 

which are the required components. 

To find the angles which their directions make with the resultant, 
we have from Equations (59), 

9" = 24° = the angle which F" makes with the resultant. 



MECHANICS OF SOLIDS. 



Y3 



and, 



9' = 14® = angle which P' makes with the resultant. 



4. -^Required the angle under which two components whose inten- 
sities are denoted by 5 and 7 should act, to give a resultant whose 

intensity is represented by 9. 

Ans. 84°, 15', 39" 

5. — From Equation (56) it appears that the resultant of two 
components applied to the same point, is greatest w^hen the angle 
made by their directions is 0°, and least when 180°. Required the 
angle under which the components should act, in order that the 
resultant may be a mean proportional between these values; and 
also the angle which the resultant makes with the greater component. 
Call P', the greater component. 

P'' 



Ans. 6 



-1 

cos 



.-1 r' 

<P = sm — . 



6. — Given a force whose intensity is denoted by 17. Required the 
two components which make with it angles of 27° and 43°. 

§ 88. — The theorem of the parallelogram of forces, just explained, 
enables us to determine by an easy graphical construction the in- 
tensity and direction of the resultant of several forces applied to the 
same point. 

Let P', P", P'", &c., be 
several forces applied to the 
same point m. Upon the 
directions of the forces, lay 
off from the point of ap- 
plication distances propor- 
tional to the intensities of 
the forces, and let these dis- 
tances represent the forces. 
From the extremity P' of 
the line mF', which rcprc- 




p\^ 



PZf^c- 



74 ELEMEITTS OF ANALYTICAL MECHANICS. 

sents the first force, draw the line P' n equal and parallel to m P" 
which represents the second, then will the line joining the extremity 
of this line and the point of application, represent the resultant of 
these two forces. From the extremity n^ draw the line nn^ equal 
and parallel to mP'" which represents the third force; mn' will 
represent the resultant of the first three forces. The construction 
being thus continued till a line be drawn equal and parallel to 
every line representing a force of the system, the resultant of the 
whole will be represented by the line, (in this instance m n")^ join- 
ing the point of application with the last extremity of the last 
line drawn. Should the line which is drawn equal and parallel to 
that wliich represents the last force, terminate in the point of appli- 
cation, the resultant will be equal to zero. 

The reason for this construction is too obvious to need expla- 
nation. 

§ 89. — If the forces still be supposed to act in the same plane, 
but upon different points of the plane, the first of Equations (49) 
takes the form, 

Tx - Xy = y. [P' (cos ^' x' - cos a' y') ], 
thus, differing from Equation (55), in giving the equation of the line 
of direction of the resultant an independent term, and showing 
that this line no longer passes through the origin. It may be con- 
structed from the above equation. 

§90. — To find the resultant in this case, by a graphical construc- 
tion, let the forces P\ 
P", P'" &c., be ap- ^, 



plied to the points m\ 



pi 

0"_ 7 f/y JE 



m", m'", &c., respec- j^^^ / \ / 

lively. Produce the / \ ^,/- -^,, 

directions of the forces *^;^' "~/^""\ /'\ 

P' and P" til] they \/ \ / \ 
meet at 0. and take 



77!!"' 




this as their common 
point of application ; 
lay off from 0, on the lines of direction, distances S and S', 



MECHANICS OF SOLIDS. 75 

proportional to the intensities of the forces P' and P", and construct 
the parallelogram S B S\ then will B represent the resultant of 
these forces. The direction of this resultant being produced till it 
meet the direction of the force P''', produced, a similar construction 
will . give the resultant of the first resultant and the force P"', 
which will be the resultant of the three forces P', P" and P'" ; 
and the same for the other forces. 

OF PAEAI.LEL FOKCES. 

§91. — If the forces act in parallel directions, 

a' = a" z= a'" = &c., 
/3' = IB" = 13"' = &c., 
y = y" = j'" = &c., 

and Equations (41) become, 

X = (P' + F" + P"' + &c.) cos a', 
r = (P' + P" + P"' + &c.) cos /3', 
Z = (P' + P" + P"' + &c.) cos 7/ ; 

these values in Equation (47) give, 

B =z ± V (^' + ^" + ^'" + &C.)2 (C0S2 a' + C0S2 /3' + C0S2 7'), 

but, 

C0S2 ct' -|_ cos2 ^' -f C0S2 7' zz: 1 ; 

hence, 

P =z P' + P'' + P"' + &c. (60) 

If some of the forces as P", P"\ act in directions opposite to 
the others, the cosines of a" and a"' will be negative while they 
have the same numerical value ; and the last equation will become 

B = P' - P" - P'" + &c. 

Whence we conclude, that the resultant of a number of -parallel 
forces is equal in intensity to the excess of the sum of the inten- 
sities of those ivhich act in one direction over the sum of the 
intensities of those which act in the opposite direction. 



76 ELEMENTS OF ANALYTICAL MECHANICS. 

§ 92. — The values of i?, X, Y and Z being substituted in Equa- 
taeas (48) give, 

(F' + F" + P'" 4- &c.) cos a' 

cos a = — " ' 



cos ^ =- D/ , z>./ , rjnr , ,-/ = COS ^\ 



F' 


+ P" 


+ P'" 


+ &c. 






{P' + P" 


+ P'" 


+ &c.) 


COS 


/3' 


F' 


+ P" 


+ F'" 


+ &c. 






{F' 


+ P" 


4- P'" 


+ &c.) 


COS 


7' 



cosc= p, ^ p., ^ p,„ ^ ^,; - = cosy 

The denominator of these expressions, being the resultant, is essen- 
tially positive ; the signs of the cosines of the angles a, h and c, 
will, therefore, depend upon the numerators; these are the compo- 
nents parallel to the three axes. 

Hence^ the resultant acts in the direction of those forces whose 
cosines are nejaiive or positive according as the sum of the former 
or latter forces is the greater. 

§93.— The forces being still parallel. Equations (42) reduce to, 

^ , ^ , {F' x' + F" x" + F'" x'" + &c.) cos /S' 

Rx cos h — B"- ^-^^ ' 



' y cos a z=z \ 



- {F'y' + F"^j" + F"'y"' + &c.) cos a' 

I 



, (P' s' + F" z" + F'" z'" -f &c.) cos a' 
(P' x' + F" x" + F'" x'" + &c.) cos y' 

j (P' y' -f- F" y" + P- y'" + &c.) cos 7' 
ic y cos c — id ^ cos 6 = ^ 

( _ (P' ^' + p'^ ^" -I- p'" z'" 4. &c.) cos /3-' 

but, 

cos ^ = cos ^', 

cos a rz: COS a', 

COS c = COS y' ; 
Substituting the second members of these last equations for the 
first in the equations immediately preceding, and transposing all the 
terms to the first member, we obtain. 



[Rx - {F'x' + F"x" + F"'x'" + &c.)] cos/S' 



s ck' ) 



\Ry - {F'y' + F"y" + F'" y'". -f &c.)] co^ 



\R z - (P' ^' + F" z" + P'" z'" + &c.)] cos oc' 



- [Pa; - (P'rr' + F"x" + P"'a;'" + &c.)] cos 7 

[Ry - {F'y' + P^'y" + F"'y'" + &c.)] cosy 
^\Rz- (P' ^' + P" z" + P'" 2"' + &c.)] cos/3 



MECHANICS OF SOLIDS, 



rr 



These equations must be satisfied, whatever may be the angles 
which the common direction of the forces makes with the co-ordinate 
axes, and this can only be done by making the co-efficients of the 
cos a', cos /3'' and cos y\ (either two of the latter being arbitrary), 
separately equal to zero. Hence, 



Rx = P'x' + P"x" + P"'x"' + &c. 
By =: P'y' + P"y" + P"'y"' + &c. 
Rz = P'z' + P"z" + P"'z"' + &c. J 



(61) 



The forces being given, the value of i?, §91, becomes known, 
and the co-ordinates a:, y, 0, are determined from the above equations ; 
these co-ordinates will obviously remain the same whatever direction 
be given to the forces, provided, they remain parallel and retain the 
same intensity and points of application, these latter elements being 
the only ones upon which the values of rr, y, z, depend. 

The point whose co-ordinates are a?, y^ 2, which is the point of 
application of the resultant, is called the centre of parallel forces^ and 
may be defined to be, that point in a system of parallel forces through 
which the resultant of the system will always pass, whatever be the 
direction of the forces^ provided, their intensities and points of appli- 
cation remain the same. 



§ 94. — Dividing each of the above Equations by i2, we shall have 
P'x' + P"x" + P"'x"' + &c. 



y = 



P' + P" 4- P'" + &c. 

P'y' + P"y" 4- P"'y"' + &c. 

P' + P" -f P'" + &c. 
P'z' -f P"z" + P"'z"' -f &c. 

P' + P" + P'" + &c. 



(62) 



Hence, either co-ordinate of the centre of a system of parallel forces 
is equal to the algebraic sum of the products which result from multi- 
plying the intensity of each force by the corresponding co-ordinate of its 
point of application, divided by the algebraic sum of the forces. 

If the points of application of the forces be in the same plane, 



78 



ELEMENTS OF ANALYTICAL MECHANICS. 



the co-ordinate plane ary, may be taken parallel to this plane, in 
wliich case 



and, 



z^' = g"' = s'"' &c. 



p, _|. pn ^ p,n j^ ^^^ 



from which it follows that the centre of parallel forces is also in this 
plane. 

If the points of application he upon the same straight line, take 
the axis of x parallel to this line ; then in addition to the above results, 
we have 

/ = 2/" = y"' = &c.; 

and, 

_ (P' + P" + P"' + &c.) y _ ,^ 
^ ~ P' + P" + F'" + &c. ~ ^ ' 

whence, the centre of parallel forces is also upon this line. 

§ 95. — If we suppose the parallel forces to be reduced to two, viz. 
P' and P"^ we may assume the axis x to pass through their points 
of application, and the plane xy to contain their directions, in which 
case, Equations (60) and (61) become, 

R = P' + P" 
Rx = P'x' + P^'x" 
2 = and y = 0. 

Multiplying the first by x\ and subtracting 
the product from the second, we obtain 

R{x - x') = P" {x" - x') . , (a) 

Multiplying the first by x" and sub- 
tracting the second fi:om the product, 
we get 

R {x" - x) = P' {x" -x') . , , . (b) 

Denoting by S' and S", the distances from the points of application 




MECHANICS OF SOLIDS. 



79 



of P' and P" to that of the resultant, which are x — x' and x" — x 
respectively, we have • 

x" -x' = S' + S" ; 

and from Equations (a) and (5), there will result 

P' : P" : R '. : S" : S' : S" + S' . . . . (63) 

If the forces act in opposite directions, then, on the supposition 
that P' is the greater, will 

R = P' - P" 
Rx = P'x' - P"x" 
= 0, 2/ = 0. 

and by a process plainly indicated by 
what precedes, 

P' : P" : P : : 5'' : S' : 5" - >S". . (64). 




From this and Proportion (63), it is 
obvious that the point of application of 
the resultant is always nearer that of the 
greater component; and that when the 

components act in the same direction, the distance between the point 
of application of the smaller component and that of the resultant, is 
less than the distance between the points of application of the com- 
ponents, while the reverse is the case when the components act in 
opposite directions. In the first case, then, the resultant is between 
the components, and in the second, the larger component is always 
between the smaller component and the resultant. 

And we conclude, generally, that the resultant of two forces which 
solicit two points of a right line in parallel directions, is equal in inten- 
sity to the sum or difference of the intensities of the components, accord- 
ing as they act in the same or opposite directions, that it always acts 
in the direction of the greater component, that its line of direction is 
contained in the plane of the components, and that the intensity of either 
component is to that of the resultant, as the distance between the point 
of application of the other component and that of the resultant, is to 
the distance between the points of application of the components. 



80 



ELEMENTS OF ANALYTICAL MECHANICS. 



7if 



§96. — Examples. — 1. The length of the Ime m' m" joining the 
points of application of two parallel forces 
acting in the same direction, is 30 feet ; the 
forces are represented by the numbers 15 
and 5. Eequired the intensity of the re- 
sultant, and its point of application. 

^ = P' + P" = 15 + 5 = 20; 
E : P' :: m" m' : m" o, 

20 : 15 :: 30 : m" o = 22,5 feet. 

A single force, therefore, whose intensity is represented by 20, applied 
at a distance from the point of application of the smaller force equal 
to 22,5 feet, will produce the same effect as the given forces applied 
at ??i" and m'. 

2. — Required the intensity and point 
of application of the resultant of two 
parallel forces, whose intensities are de- 
noted by the numbers 11 and 3, and 
which solicit the extremities of a right 
line w^hose length is 16 feet in opposite 
directions. 



77/" 




P rr P' - P" =. 11 - 3 = 8, 



P' - P" : P' : : m" m' ; m" o = 



P' . m" w! 



= 22 feet. 



P' - P' 

3. — Given the length of a line whose extremities are solicited in 
the same direction by two forces, the intensities of which differ by 
the n^^ part of that of the smaller. Required the distance of the 
point of application of the resultant from the middle of the line. 
Let 2 I, denote the length of the line. Then, by the conditions, 

^ 4- 1 \ ^„ 

2n + 1 ^„ 



P' = E" + —P" = ( 






R = (!L±1) P" + P" = 

\ n / 

\ n y 



2nl 



2n + 1 



CO =: I 



2nl 



1 



2n + I 2/1+1 



I. 



MECHANICS OF SOLIDS, 



81 



§97. — The rule at the close of §05, enables us to determine by a 
very easy graphical construction, the position and point of application 
of the resultant of a number of parallel forces, whose directions, 
intensities, and points of application are given. 

Let P, P\ P", F", and P% 
be several forces applied to the 
material points w, m\ m'\ m'", 
and m% in parallel directions. 
Join the points m and m' by a 
straight line, and divide this line 
at the point o, in the inverse 
ratio of the intensities of the 
forces P and P' ; join the points 
and m" by the straight line 
om"^ and divide this lihe at o\ 

in the inverse ratio of the sum of the first two forces and the force 
P" J and continue this construction till the last point m^^ is included, 
then will the last point of division be the point of application of the 
resultant, through which its direction may be drawn parallel to that 
of the forces. The intensity of the resultant will be equal to the 
algebraic sum of the intensities of the forces. 

The position of the point o will result from the proportion 




P + P' : P' :: m m! 
that of o' from 

that of o" from 

P ^ P' -\- P" - P'" : - P'" 

and finally, that of o'" from 



mo z=z 



P' 



P + P' 



om" : oo' = 



0' m' 



P" .om' 



P-\-P' + P' 



P + P'+ P" -W' 



o"m' 



p^2J'^P"—P"'Jf.p^ 



S§ ELEMENTS OF ANALYTICAL MECHANICS. 



OF COUPLES. 

§98. — When two forces P' and P" act in opposite directions, the 
distance of the point o, at which the resultant 
is applied, from the point m', at which the 
component P' is applied, is found from the 
formula 



m' 




and if the components P' and P" become 

equal, the distance m' o will be infinite, and 

the resultant, zero. In other words, the forces 

will have no resultant, and their joint effect 

will be to turn the line m" m\ about some "point between the points 

of application. 

The forces in this case act in opposite directions, are equal, but 
not immediately opposed. To such forces the term couple is applied. 
A couple having no single resultant, their action cannot be compared 
to that of a single force. 

§99. — The analytical condition. Equation (46), expressive of the 
existence of a single resultant in any system of forces, will obviously 
be fulfilled, when 

X = 0, F = 0, and Z = 0. 

But this may arise from the parallel grou2:)s of forces whose sums 
are denoted by X, F, and Z, reducing each to a couple. These three 
couples may easily be reduced by composition to a single couple, 
beyond which, no further reduction can be made. It is, therefore, a 
failing case of the general analytical condition referred to. 

WORK OF THE RESULT Al^^T AND OF ITS C0:MP0NENTS. 

§ 100. — We have seen that when the resultant of several forces 
is introduced as an additional force with its direction reversed, it 
will hold its components in equilibrio. Denoting the intensity of 



MECHANICS OF SOLIDS. 83 

the resultant by i?, and the projection of its virtual velocity by 
(Jr, we have from Equation (29), 

- R5r -{• PJp -\- P'Jp' + F".8p" + &c. = 0, 



BSr = F.dp + P' 8p' + P" Sp" + &c., .... (65) 

in which P, P' P'\ &c. are the components, and ^ p^ dp' §p'\ &c. 
the projections of their virtual velocities. 

§101. — Now, the displacement by which Equation (29) was de- 
duced, was entirely arbitrary ; it may, therefore, be made to conform 
in all respects to that which would be produced by the components 
P, P', &ZG., acting without the opposition of the force equal and 
contrary to their resultant; and writing dr for Sr, dp for Sp, &c., 
Equation (65) will become 

Edr =Pdp + P'dp' + P"dp" + (fee, . . . (66) 

and integrating, 

fEdr = fPdp-h fP'dp' + fP"dp" + &c., . . (67) 

in which R^ P, P\ &c. may be constant or functions of r, p^ p\ &c., 
respectively. 

From Equations {^^) and (67), it appears that the quantity of 
work of the resultant of several forces is equal to the algebraic sum 
of the quantities of work of its -components. 

Again, replacing P^p^ P' 6pj\ &c. in Equation {Q^)^ by their values 
in Equation (31), and writing dr for (Jr, dp for 6 p^ &;c., we find, 

fRdr :=z flP.cosa.dxH- flPcos(3.dij -i-fllP.cosy.dz, - - (68) 

in which R may be constant or a function of r; P, constant or a 
function of x, y, z, &c. 

If the forces be in cquilibrio, then will P = 0, and, 

2P. cosa.tf.r + 2P. coh [3. dy -f- IP. cos y.dz — 0. • • (69) 



84: 



ELEMENTS OF ANALYTICAL MECHANICS. 



MOMENTS. 



§ 102. — If the forces act in the same plane, or in parallel planes, 
the axis of z may be assumed perpendicular thereto, in which case, 

cos7= cos^'=&c.=:0; cos5=:sinaj cos/3'=sina'; cos/3"=z:sina" &c. 

and the first of Equations (42) becomes, 

i2 (a; . sin a — y . cos a) = 2 P' . {x' sin a' — y' cos a') • . (70) 

Denote by H^ the length 
of the line M A^ drawn from 
the point of application M 
of jS, perpendicular to the 
axis ^, and by 9, the angle 
which this line makes with 
the axis x. Multiplying and 
dividing the first member of 
the above equation by H^ and 
reducing by the relations, 




~X 



-^ = cos (p 



and 



H 



= sm (p ; 



sin a cos 9 — cos a sin 9 = sin (a — 9), 
there will result, 

R {x sin a — y cos a) =\R. H . sin (a — 9). 

But if a line A 0, be drawn from the point A^ perpendicular to 
the line of direction of i?, and its length be denoted by K^ then will 

H. sin (a — ^) =: K\ 

which, in the above equation, gives 

R{x .%ma — y cos a) — R . K. (71) 



In the same way, denoting the lengths of the perpendiculars 
drawn from the points in which the axis 0, pierces the planes of 



MECHANICS OF SOLIDS. 



85 




the forces P', P", &;c., to their respective lines of direction by 
Tc'y k", &c., will, 

2 P'. {x' sin a' - y' cos a') = 2 P'. A;' ; • • • (72) 

and Equation (70) may be written, 

R.K^^P'.k'. (73) 

§ 103.— The lines K, Jc\ 
&;c., are called the lever 
arms of the forces P, P', 
&c., and Equation (73) 
shows that the quantity 
of work of the resultant of 
several forces acting in 
parallel planes^ and through 
a distance equal to its lever 
arm, is equal to the alge- 
braic sum of the quanti- 
ties of work of its compo- 
nents acting through distances equal to their respective lever arms. 

§ 104. — The quantity of work measured by the product of the inten- 
sity of a force by its lever arm, is called the moment of the force. 

A line perpendicular to the plane of the force and its lever arm, 
and through the extremity of the latter most remote from the line 
of direction of the force, is called the moment axis. 

The point in which the moment axis pierces the plane of the 
force and lever arm, is called the centre of moments. 

A line through the centre of moments and oblique to the plane 
of the force and its lever arm, is called a component axis. 

The moment of the resultant of several forces, is called the result- 
ant moment. 

The moments of the several components, are called component 
moments — the corresponding axes being called respectively resultant 
and component axes. 

§ 105. — If the moment axis be fixed, the virtual velocities will 
be arcs of circles. 



86 ELEMEl^-TS OF AITALTTICAL MECHANICS. 

Let M be the point of application 
of the force R ; MR, its direction ; ' ~' 

A, its centre of moments ; MN its 
virtual velocity; MO, the projection 
of the latter, and AD = K, the 
lever arm. 

The virtual moment, and therefore the elementary quantity of 
work of R will be. 

R,MO, 

Denote the space described at the unit's distance by ds^, then 




will. 



M]sr= ir.ds^, 

MO = H,ds,.cosNMO 



but because AM and AD are respectively perpendicular to MN 
and MO, the angle OMJSf is equal to the angle DAM, and 



cos NM = 






which substituted above, gives 

MO 



K.d^ 



and multiplying by R, 



R.MO = R.K.ds.. 



That is to say, the elementary quantity of work performed by a 
force while its point of application is constrained to turn about its 
moment axis, is equal to the moment of the force multiplied by the 
differential of the arc described at the unit's distance from this axis. 



106. — Multiplying both members of Equation (73) by ds^^ we 
R.K.ds. = l,P'.k'.ds.. 



get, 

and integrating, 



SR.K.ds, = f^P'.k'ds^ 



(74) 



MECHAK-JCS OF SOLIDS. 87 

§ 107. — The effect of a force acting at the end of its lever arm, 
is to produce rotation about the other end as the centre of moments, 
supposed fixed ; the resistance at the centre of moments is equal 
and contrary to the action of the force ; the action of the force and 
this reaction form, therefore, a couple^ and the lever arm of the force 
is also called the lever arm of the couple. 

The moments of the forces which urge a body to turn in opposite 
directions about any assumed axis must have contrary signs. 

The sign of P' h\ or its equal P' sin a! .x' — P' cos a,' . y\ depends 
upon the angles which the direction of the force makes with the 
axes and upon the signs and relative values of the co-ordinates of 
the point of application. 

Let the angles which the direction of any force makes with the 
co-ordinate axes be estimated from the positive side of the origin ; 
then, if the angles which this direction makes with both axes 
be acute, and the point of application lie in the first angle, 
P'^WLo! .x' and P' qosol' .y\ will be positive, and if the first of 
these products exceed the second, the moment will be positive ; but 
if the latter be the greater, the moment will be negative. 

In the first case, the direction of P' will pierce the plane x' z' 
on the side of x' positive and the plane y' z' on the side of y' 
negative, while in the second case the reverse will be true. Now 
conceive the lever arm and line of direction of the force to rotate 
together about the axis 2, the various positions assumed by them 
in one revolution will indicate the circumstances which produce a 
positive moment with respect to the axis z. If in any one posi- \ 
tion either the action of the force be reversed, or the line of di- 
rection be transferred to the opposite side of the axis, a negative 
moment with respect to z will result. The same of the axes x and y. 

§ 108. — The forces having any direction, each force may be replaced 
by its three components, parallel respectively to the rectangular axes 
2;, y, z. The components parallel to the axis 2, can, § 102, have no in- 
fluence to produce rotation about that line, and the effect of all the forces 
in this respect, will be the same as that of their components parallel to 
the axes x and y. But these act in planes at right angles to the 



88 ELEMENTS OF ANALYTICAL MECHANICS. 

axis z, which axis being taken as their moment axis, the effects of 
these components may be computed by Equations (70), (73), (74) ; 
and by reference to Equations (42) and (72), it will be. seen that 
the quantities Z, M^ iV, are the algebraic sums of the moments of 
all the forces in reference to the axes 0, y, and x^ respectively. 

COMPOSITION AND RESOLUTION OF MOMENTS. 

§ 109. — The forces being supposed to act in any directions whatever, 
join the point of application of the resultant R and the origin by 
a right line, and denote its length by H. Multiply and divide each 
of the Equations (44) by H, and reduce by the relations, 

-Jl = cos I, 

^ ^ cos i, 

— = COS £, 

in which ^, | and s, denote the angles which the line H makes 
•with the axes ar, y and z, respectively ; then will 

R. H . (cos h . cos ^ — cos a . cos |) = Z, ^ 

R . H . (cos a . cos s — cos c . cos ^) = AI, y . . . (75) 

R . H . (cos c . cos f — cos b . cos s) = iV. J 

Squaring each of these Equations and adding, we find 

f cos^ h . cos^ ^ — 2 cos b . cos a . cos ^ . cos | + cos^ a . cos^ | 
jR^ . ff2 ^ _|_ cQs2 ^ ^ QQ^2 g — 2 cos a . cos c . cos s . cos ^ + cos^ c . cos^ ^ 

I -f-COS^ C . COS^ I — 2 cos 5 . cos C . COS I . COS s + cos^ b . cos^ s 

r= Z2 + iP + iy2 (76) 

But 

cos^ a + cos^ 6 4- cos^ c = 1, (77) 

cos^ ^ + cos^ I 4- cos^ ^ = 1, C^S) 

cos a . cos ^ 4- cos b . cos ^ + cos c . cos s = cos 9, . (79) 



MECHANICS OF SOLIDS. 89 

the angle 9, being that made by the Ihie H^ with the direction of 
the resultant. 

Collecting the co-efficients of cos^ a^ cos^ 6, cos^ c, and reducing 
by the following relations, deduced from Equation (78) ; viz. : 

COS^ S + C0S2 1 = 1— C0S2 ^, 
COS^ ^ -f C0S2 6 = 1— C0S2 f , 
C0S2 I _^ (.Qs2 ^ _ 1 _ cos2 £, 

we find, 

m.H'^.\\- (cos a . COS ^ + cos 6 . cos f + cos c . cos 5)2] z=zU--^M^^N'^ ; 

from Equation (T9), 

1 — (cos a . cos ^ + cos b . cos ^ + cos c . cos sy z= 1 — cos^ 9 =1 sin^ 9 ; 

which reduces the above to 

E^.H"^. sm2 9 = Z2 + J/2 _^ iV 2. 

But JT^ ^ sin2 (p is the square of the perpendicular drawn from the 
origin to the direction of the resultant ; it is, therefore, the square 
of the lever arm of the resultant referred to the origin as a centre 
of moments. Denoting this lever arm by K^ we have, after taking 
the square root, 

R.K ^ V Z2 + J/2 + iV2 (80) 

That is to say, the resultant moment of any system of forces is equal 
to the square root of the sum of the squares of the su7ns of the com- 
ponent moments^ taken in reference to any three rectangular axes through 
the point assumed as the centre of moments. 

§ 110. — Dividing the first of Equations (T5), by Equation (80), 
we find, 

H (cos b . cos ^ — cos a . cos ^) L 



■^ -y/V- + J/2 + iV2 * 

The effect of a force is, §77, indepenrlent of the position of its 
point of application, provided it be taken on the line of direction. 
Let the point of application of i?, be taken at the extremity of its 



90 ELEMEN-TS OF ANALYTICAL MECHAI^ICS. 

lever arm, then ^Yill H coincide with and be equal in length to K\ 
^ and f will become the angles which the lever arm makes with the 
axes X and y, respectively, and the well known relation obtained 
from the formulas for the transformation of co-ordinates from one 
set of rectangular axes to another, will give 

cos A = cos h . cos ^ — cos a . cos f . 

in which A is the angle the resultant axis makes with the axis z ; 
whence,* 

cos A = —— ^ (81) 

In the same way, denoting by B and C the angles which the* 
moment axis of R makes with the co-ordinate axes y and x respec- 
tively, will, 

M 
cos B = ^ (82) 

cos C = ^ =. (S3) 

whence we conclude that, the cosine of the angle which the resultant 
axis maJces with any assumed line is equal to the sum of the moments 
of the forces in reference to this line taken as a component axis 
divided by the resultant moment. 

§111.— Multiplying Equation (81) by Equation (80), there will 
result, 

R.K. cosA=L (84) 

which shows that the component moment of any system of forces in 
reference to any oblique axis is equal to the product of the resultant 
moment of the system into the cosine of the angle between the resultant 
and component axes. 

For the same system of forces and the same centre of moments, 
it is obvious that R and K will be constant ; whence, Equation (80), 
the sum of the squares of the sums of the moments in reference 
*See Appendix No. 1. 



MECHANICS OF SOLIDS. 91 

to any three rectangular axes through the centre of moments, taken 
as component axes is a constant quantity. Also, since the axis z 
may have an infinite number of positions and still satisfy the con- 
dition of making equal angles with the resultant axis, we see, 
Equation (84), that the sum of the moments of the forces in reference 
to all component axes which make equal angles with the resultant 
axis will be constant. 

§ 112.— Denote by &', &", &'", the angles which any component 
axis makes with the co-ordinate axes s, y and x, respectively, and 
by the angle which the component and resultant axes make with 
each other, then will 

coso = cos^ . cos &' + cos B . cos ^'' 4- cos C. cos ^'", 

multiplying both members by B.Il, we have 

B.K.cosd =B,E.cosA.cos&' -{-E.K.cosBcos6" + B.IC.COS C.cos&"\ 
But, Equation (84), 

B . K . cos A =: L., 
B . K. cos B = M, 

B . K. cos C z=: N; 

which substituted above, gives 

B.K. cos =: L. COS ti' -f M. cos 6" + ^^. cos &"' . . • (85) 

That is to say, the component moment in reference to any assumed com- 
ponent axis, is equal to the sum of the i)roducts arising frdm multiplying 
the sum of the moments in reference to the co-ordinate axes, by the 
cosines of the angles which the direction of the component axis 7/iakes 
with these co-ordinate axes, respectively. 

TRANSLATION OF EQUATIONS {A) AND (B). 

§ 113. — ICquatioiis (A) and {B) may now be translated. They express 
the conditions of equilibrium of a system of forces acting in various 
directions and upon dilFcrcnt points of a solid body. These condi- 
tions are six in number ; viz. : 



92 ELEMEN'TS OF AE"ALYTICAL MECHANICS. 

1. — The algebraic sum of the components of the forces in each of 
any three rectangular directions 7nust be separately equal to zero ; 

2. — The algebraic sum of the moments of the forces taken in refer- 
ence to each of three rectangular axes drawn through any assumed 
centre of moments^ must he separately equal to zero. 

If the extraneous forces be in equilibrio, the terms which measure 
the forces of inertia will disappear, and these conditions of equilibrium 
will be expressed by 



2 P . cos a z= 0, 
2 P cos /3 = 0, 
2 P. cos 7 = 0;^ 



2 P . (a;' cos ^ — y' cos a) = 0,' 
2 P . {z'. cos a — x' cos y) = 0, 
'S, P . (y' cos y — z' cos /3) = 0. 



{AY 



{BY 



The above conditions, which relate to the most general action 
of a system of forces, are qualified by restrictions imposed upon 
the state of the body. 

§114. — If the body contain 2. fixed pointy the origin of the mova- 
ble co-ordinates, in Equation (40), may be taken at this point ; in 
which case we shall have, 

Zx, = 0, 
h. = 0, 

and it will only be necessary that the forces satisfy Equations 
(B), these being the co-efficients of the indeterminate quantities that 
do not reduce to zero. Hence, in the case of a fixed point, the 
sum of the moments of the forces, taken in reference to each of three 
rectangular axes, passing through the point, must separately reduce to 
zero. 

Should the system contain two fixed points, one of the axes, as 



MECHAN"ICS OF SOLIDS. 93 

that of X, may be assumed to coincide with the line joining these 
points, in which case, there ^Yill result in Equation (40), 

Sxj =z 0, 6"(p = 0, 

Sz, = 0, 

and it will only be necessary that the forces satisfy the last Equa- 
tion in group (£) ; or that the sum of ike moments of the forces in 
reference to the line joining the fixed points^ reduce to zero. 

If the system be free to slide along this line, S x^ will not reduce 
to zero, and it will be necessary that its co-efficient, in Equation 
(40), reduce to zero ; or that the algebraic sum of the components of 
the given forces parallel to the line joining the fixed points, also reduce 
to zero. 

If three points of the system be constrained to remain in a 
fixed plane, one of the co-ordinate planes, as that of xy, may be 
assumed parallel to this plane; in which case, 

hz, = 0, 
6zi = 0, 
H-0; 

and the forces must satisfy the first and second of Equations {A), 
and the first of {B) ', that is, the algebraic sum of the components 
of the given forces parallel to each of two rectangular axes parallel to 
the given plane, must separately reduce to zero, and the sum of the 
moments in reference to an axis perpendicular to this plane must reduce 
to zero. 

CENTEE OF GEAVITY. 

§115. — Gravity is the name given to that force which urges all 
bodies towards the centre of the earth. This force acts upon every 
particle of matter. Every body may, therefore, be regarded as 
subjected to the action of a system of forces whose number is equal 
to the number of its particles, and whose points of application have, 
with respect to any system of axes, the same co-ordinates as these 
particles. 



94 ELEMENTS OF ANALYTICAL MECHANICS. 

The iceight of a body is the resultant of this system, or the 
resultant of all the forces of gravity which act upon it, and is equal, 
in intensity, but directly opposed to the force which is just sufficient 
to support the body. 

The direction of the force of gravity is perpendicular to the 
earth's surface. The earth is an oblate spheroid, of small eccentri- 
city, whose mean radius is nearly four thousand miles; hence, as the 
directions of the force of gravity converge towards the centre, it is 
obvious that these directions, when they appertain to particles of 
the same body of ordinary magnitude, are sensibly parallel, since 
the linear dimensions of such bodies may be neglected, in compari- 
son with any radius of curvature of the earth. 

The centre of such a system of forces is determined by Equa- 
tions (62), §94, which are 

4- &c. 



Vi = 





P' + 


P" 


-\- P'" 


+ &c. 


p 


y + p 


'T 


+ P'" 


y'" + &c. 




p' + 


P" 


+ P'" 


4-. &c. 


p 


'z' + P 


"z" 


+ P'" 


z'" + &c. 



p, ^_ p„ ^ pl„ ^ (.^^ 



(86) 



in which x^ y^ z^, are the co-ordinates of the centre ; P\ P", &c., 
the forces arising from the action of the force of gravity, that is, 
the weights of the elementary masses m', m'\ &c., of which the 
co-ordinates are respectively x' y' z\ x" y" z", &c. 

This centre is called the centre of gravity. From the values of 
its co-ordinates, Equations (86), it is apparent that the position of 
this point is independent of the direction of the force of gravity in 
reference to any assumed line of the body; and the centre of gravity 
of a body may be defined to be tliat point through which its weight 
ahoays passes in whatever way the body may be turned in regard to 
the direction of the force of gravity. 

Tlie values of P', P", &;c., benig regarded as the weights iv', w'\ 
&c., of the elementary masses w', m'\ &c., we have. Equation (1), 

P' = w' =^ m'g'; P" = w" = m" g" ; P'" =z iv'" = m'" g'" -, (fee. 



MECHANICS OF SOLIDS. 



95 



and, Equations (86), 



m'g'x' + m"g"x" + m'" g'" x'" + &c. 



ys = 





m' 


9' 


+ 


m" g" 


+ 


m"'g'" 


+ &c. 


m 


9'y' 


+ 


m 


"9"y" 


+ 


m"'g" 


>"' + &c. 




'in' 


9' 


+ 


m"g" 


+ 


m"'g"' 


+ &c. 


m' 


9'^' 


+ 


m 


' g" z" 


+ 


m'" g'" z'" + &c. 



m' g' + m" g" + m'" g'" + &c. 



(87) 



§116. — It will be shown by a process to be given in the proper 
place, that the intensity of the force of gravity varies inversely as 
the square of the distance from the centre of the earth. The distance 
from the surface to the centre of the earth is nearly four thousand 
miles ; a change of half a mile in the distance at the surface would, 
therefore, only cause a change of one four-thousandth part of its 
entire amount in the force of gravity; and hence, within the limits 
of bodies whose centres of gravity it may be desirable in practice to 
determine, the change would be inappreciable. Assuming, then, the 
force of gravity at the same place as constant, Equations (87), 
become * 

m' x' + m" x" + m'" x'" + &c. 



y, = 





m' + m" 


+ 


m'" 


+ &c. 


Tfl' 


y' Jr m" y" + 


m" 


y'" + &c. 




m' -f m" 


+ 


m'" 


+ &c. 


m' 


z' + m" z" 


+ 


m'" 


z'" + &c. . 



&c. 



(88) 



from which it appears, that when the action of the force of gravity 
is constant throughout any collection of particles, the position of the 
centre of gravity is independent of the intensity of the force. 

§ 117. — Substituting the value of the masses, given in Equation (1)', 
there will result, 

v' d'x' + v"d" x" + v"'d"'x"' + (fcc. 



v'd' + v"d" + v"'d"' + &c. 
v'd'y' + v"d"y" + v'" d'" y'" + &c. 

v'd' + v" d''~\- v"'d"' + &c. ' 
v'd' z' + v" d" z" + v '" d'" z'" + &c. 

v' d' + v" d" + v'" d'" + &c. ' 



(89) 



96 



ELEMENTS OF ANALYTICAL MECHANICS. 



and if the elements be of homogenous density throughout, we shall 
have, 

d' = d" =: d'" = &c. ; 

and Equations (89) become, 



y, = 







v' 


+ 


v" 


-t- 


v'" 


+ &c. 


v' 


y' 


+ 


v"y" -\- 


v'" y'" + &c. 






v' 


+ 


v" 


+ 


v'" 


+ &c. 


v' 


z' 


+ 


v" z' 


' + 


v'" z'" -f &c.. 



v' + v" + «<"' + &c. 



(90) 



whence it follows, that in all homogeneous bodies, the position of 
the centre of gravity is independent of the density, provided the 
intensity of gravity is the same throughout. 

§ 118. — Employing the character 2, in its usual signification. Equa- 
tions (90), may be written, 

2 {yx) 



""'- l{v) 



y^ = 



2(z;y) 



2(. 
2 (v 2) . 



' > 



(91) 



and if the system be so united as to be continuous, 

y„" X. dV 



3/; = 



Jv" y- 



dV 



Jv" ^ ' 



dV 



(92) 



§119. — If the collection be divided symmetrically by the plane 
ry, then will 

2(v2r) =0, 



MECHANICS OF SOLIDS. 



97 



and, therefore, 



hence, the centre of gravity will lie in this plane. 

If, at the same time, the collection of elements be symmetrically 
divided by the plane x 0, we shall have, 

2 {yy) = 0, 
2// = ; 

the collection of elements will be symmetrically disposed about the 
axis Xj and the centre of gravity will be on that line. 

Although it is always true, that the centre of gravity will lie in 
a plane or line that divides a homogeneous collection of particles 
symmetrically; yet, the reverse, it is obvious, is not always true, 
viz. : that the collection will be symmetrically divided by a plane or 
line that may contain the centre of gravity. 

Equations (92) are employed to determine the centres of gravity 
of all geometrical figures. 

THE CEJSTTEE OF GRAVITY OF LINES. 



§ 120. — Let s represent the entire length of an arc of any curve, 
whose centre of gravity is to be found, and of which the co-erdi- 
nates of the extremities are x\ y\ z\ and x" ^ y'\ z" . 

To be applicable to this general case of a curve, included within 
the given limits, Equations (92) become 



f xdx.^ 



1 + 



dy' 



+ 



d x^ 



y, 





s 






c 


ydx.yj \ + 


dy^ 
dx^ 


^ dx-^ 




s 






c 


zdx.sj \ -f- 


dy^ 
dx' 


dz' 
"^ dx^ 



(93) 



08 



ELEMENTS OF ANALYTICAL MECHANICS. 



in which 



fj' ^ V 



1 + 









(94) 



Example 1. — Find the positioii of 
line. Let, 

y = a X + P, 
z = a'x + /3', 

be the equations of the 
line. 

Differentiating, substi- 
tuting in Equations (94) 
and (93), integrating be- 
tween the proper limits, 
and reducing, there will 
result, 



centre of gravity of a right 




x' + x" 

X. — •> 



y,^'^-^^,, 

, = ':^^-^ + ^', 



which are the co-ordinates of the middle point of the line ; x' y' z' 
and x" y" z'\ being those of its extremities ; whence we conclude 
that the centre of gravity of a straight line is at its middle point. 



Example 2. — Find the centre of gravity of the perimeter of a polygon. 

This may be done, according to Equations (90), by taking the sum 
of the products which result from multiplying the length of each side 
by the co-ordinate of its middle point, and dividing this sum by the 
length of the perimeter of the polygon. Or by construction, as fol- 
lows : 

The weights of the several sides of the polygon constitute a system 
of parallel forces, whose points of application are the centres of 
gravity of the sides. The sides being of homogeneous density, their 
weights are proportional to their lengths. Hence, to find the centre 



MECHANICS OF SOLIDS. 



99 



of gravity of the entire polygon, join the middle points of any two 
of the sides by a right line, and divide this line in the inverse ratio 
of the lengths of the adjacent sides, the point of division will, § 97, 
be the centre of gravity of these two sides ; next, join this point 
with the middle of a third side by a straight line, and divide this 
line ill the inverse ratio of the sum of first two sides, and this third 
side, the point of division will be the centre of gravity of the three 
sides. Continue this process till all the sides be taken, and the last 
point of division will be the centre of gravity of the polygon. 

Find the position of the centre of gravity of a plane curve. 
Assume the plane oi xy to coincide with the plane of the curve, 
in which case, 

and Equations (93) and (94) become, 



f xd.sj 



1 + 



dy^ 



y, = 



fl ydx^ 



1 + 



dx^ 



(95) 



C'-^h^- 



(96) 



Example 3. — Find the centre of gravity of a circular arc. 

Take the origin at the centre of curvature, and the axis of y 
passing through the middle point of the arc. The equation of the 
curve is, 



2/^- 


a2 


— X 


dy 
dx ~ 


— 


X 



whence. 



which substituted in Equations (95), 




100 ELEMENTS OF ANALYTICAL MECHANICS, 
will give on reduction, 



X, = 0, 



_ ^ {^' + ^") ♦ 



and denoting the chord of the arc by c ■= x' -{■ x'\ 



X, = 0, 
ac 



whence we conclude that the centre of gravity of a circular arc %s 
an a line drawn through the centre of curvature and its middle pointy 
and at a distance from the centre equal to a fourth proportional to 
the arc^ radius and chord. 

Example 4. — Find the centre of gravity of the arc of a cycloid. 

The radius of the generating circle being a, the differential equa- 
tion of the curve is, 



dx 



y 'dy 



-y/^ay — y'^ 



(a) 




the origin being at A^ and 

AB being the axis of x. ji 

Transfer the origin to C, 
and denote by x\ y' the new 

co-ordinates, the former being estimated in the direction CD, and the 
latter in the direction DA. Then will 



y =r 2a - x\ 



x ^^ att — y 



and therefore. 



2a 



dx dy' 

dy~dx' ~ ^%ax' — x'-- 



{ay 



MECHANICS OF SOLIDS. 101 

this, in Equations (96) and (95), gives, omitting the accent on the 
variables, 



=/:: 



. 2a 
ax 

X 



I ,, xdx 



2a 



y, = ^ 

Integrating the first two equations between the limits indicated, 
and substituting the value of 5, deduced from the first, in the second, 
we have, 

s = 2 V2a(V^" - V^'l 

_ 1 ^x"^ - Vx'\ 

and from the third equation we have, after integrating by parts, 
sy^ = 2^2a{yy/x — f -x/^dy) ; 

substituting the value of dy, obtained from Equation (a)', and re- 
ducing, there will result. 



sy^ — 2 ^2 a (y ^ X — f^2a — x .dx), 
and taking the integral between the indicated limits, 

sy, = 2 ^/2^ [y(V^' - V^) + | (2a - x^yl - |(2a - x^)i]; 
hence, replacing s by its value, and dividing, 

3 3 

, - (2a-x"y - {2a - rr') ^ 

Supposing the arc to begin at C, we have, 

x' = 0, 

and, 

X — ^x" 

3 y a; L. _j 



102 



ELEMENTS OF ANALYTICAL MECHANICS. 



If the entire semi-arc from C to A be taken, these values become. 



= fa, 



Taking the entire arc A C £, the curve will be symmetrical with res- 
pect to the axis of x', and therefore, 

2/; = 0; 
hence, the centre of gravity of the arc of the cycloid, generated by one 
entire revolution of the generating circle, is on the line which divides 
the curve symmetrically, and at a distance from the summit of the curve 
equal to one-third of its heii 



THE CENTEE OF GEAYITY OF SURFACES. 

§ 121. — Let L = 0, he the equation of any surface ; L being a 
function of xyz\ then will dxdy, be the projection of an element 
of this surface, whose co-ordinates are xy z, upon the plane x y ; and 
if ^" denote the angle which a plane tangent to the surface at the 
same point makes with the plane xy, the value of the element itself 
will be 



dx .dy 

cos &" ' 

But the angle which a plane 
makes with the co-ordinate 
plane x y, is equal to the 
angle which the normal to 
the plane makes with the 
axis 2, and, therefore. 



cos 



= ± 



dL 

dz 







(97) 



MECHANICS OF SOLIDS. 
and hence, in Equations (92), omitting the double sign, 
dV=^dx'dy'W,, , . . 
and those Equations become, 



J y J X 



J M J X 



J y J X 



vj . X .dx .dy 



w .ydx . dy 



w .z .dx .dy 



in which, 



s = V = ,, „w.dx.dy; 



w being a function of x^ y, z. 

If the surface be plane, the 
plane oi xy may be taken in the 
surface, in which case, 

w> = 1, 



come. 



y, 



Jy"L" dy 


xdx 


s 

jy"jx" dx 


ydy 


s 


.dy, 



108 



(98) 



(99) 



(100) 




and Equations (99), and (100), be- ^ — ^t j} -^, — ^ 



(101) 



(102) 



in which the integral is to be taken first with respect to y, and 



104 



ELEMENTS OF ANALYTICAL MECHANICS. 



between the limits y" = P m" and y' = P m' -, then in respect to x^ 
between the limits x" = AP'\ and x' = AP', Hence 



/ 



Vi 



." [y" -y').xdx 



\L"W"'-y"')d, 



(103) 






L 



y') «f« 



(104) 



y' and y'', denoting running co-ordinates, which may be either loots 
of the same equation, resulting from the same value of x^ or they 
may belong to two distinct functions of x^ the value of x being the 
same in each. For instance, if 

F {xy) = 0, 

be the equation of the curve n' m" n" m\ it is obvious that between 
the limits x" = A P" and x' — A P\ every value of .t, as A P, 
must give two values for y, viz.: y" = Pm" and y' = P m\ Or if 



Fixy) = 0, 
F' {xy) = 0, 

be the equations of two distinct 
curves m" n" and ml n\ referred 
to the same origin A^ then will 
y" and if result from these 
functions separately, when the 
same value is given to a; in 
each. 




Example 1. — Required the position of the centre of gravity of the 
area of a triangle. 



MECHANICS OF SOLIDS, 



105 



Let ABC, be the triangle. 
Assume the origin of co-ordi- 
nates at one of the angles A, 
and draw the axis y parallel to 
the opposite side B C. Denote 
the distance A P by x'^ and 
suppose, 

y" = ax, 



to be the equatioi'% of the sides A C and AB, respectively, then 
will 

y" - y' = (« - 5) ^, 

y"2 _ y'2 ^ (^2 _ 52j ^2^ 

and, 




/ [a ~ b) x^ do. 
I (a — b) X dx 



3^ 



2// = 



^ f\o? - 62) X-^dx 

J X' 

y- . . 

(a — b) X d X 
xr 



2 (g + 5) X \ 

y 2 



whence we conclude, that the centre of gravity of a triangle is on a 
line drawn from any one of the angles to the middle of the opposite 
side, and at a distance from this angle equal to two- thirds of the line 
thus drawn. 



Example 2. — Find the centre of gravity of the area of any polygon. 

From any one of the angles 
as ^, of the polygon, draw lines 
to all the other angles except 
those which arc adjacent on either 
side ; the polygon will thus be 
divided into triangles. Eind by 
the rule just given, the centre of 
gravity of each of the triangles; 




106 ELEMENTS OF ANALYTICAL MECHANICS. 

join any two of these centres by a right line, and divide this line in 
the inverse ratio of the areas of the triangles to which these centres 
belong ; the point of division will be the centre of gravity of these 
two triangles. Join, by a straight line, this centre with the centre of 
gravity of a third triangle, and divide . this line in the inverse ratio 
of the sum of the areas of the first two triangles and of the third, this 
point of division v>411 be the centre of gravity of the three triangles. 
Continue this process till all the triangles be embraced by it, and the 
last point of division will be the centre of gravity of the polygon ; 
the reasons for the rule being the same as those given for the deter- 
mination of the centre of gravity of the perimeter of a polygon, it 
being only necessary to substitute the areas of the triangles for the 
lengths of the sides. 

Example 3. — Determine the position of the centre of gravity of a 
circular sector. 

The centre of gravity of the sec- 
tor will be on the radius drawn to 
the middle point of the arc, since this 
radius divides the sector symmetri- 
cally. Conceive the sector CAB, to 
be divided into an indefinite number 
of elementary sectors ; each one of 
these may be regarded as a triangle 
whose centre of gravity is at a dis- 
tance from the centre (7, equal to 

two-thirds of the radius. If, therefore, from this centre an arc be 
described with a radius equal to two-thirds the radius of the sector, 
this arc will be the locus of the centres of gravity of all the 
elementary sectors ; and for reasons already explained, the centre of 
gravity of the entire sector will be the same as that of the portion 
of this arc which is included between the extreme radii of the sector. 
Hence, calling r the radius of the sector, a and c its arc and chord 
respectively, and x^ the distance of the centre of gravity from the 
centre (7, we have, 

f r . Jc 2 r .€ 




MECHANICS OF SOLIDS. 



lor 



The centre of gravity of a circular sector is therefore on the radius 
drawn to the middle point of the arc of the sector^ and at a distance 
from the centre of curvature equal to tvjo-thirds of a fourth propor- 
tional to the arc^ chord and radius of the sector. 

Example 4. — Find the centre of gravity of a circular segment. 

Assume the origin at the centre (7, 
and take the axis x passing through the 
middle point of the arc, the centre of 
gravity in question will be on this axis, 
and, therefore, 

y, = 0. 

Let A B HA be the segment, and 

y z= zt. -\/ a'^ — a;2, 

the equation of the circle, the origin being 
at the centre (7, then will 




-/a^ 



y' — — -y/ a'^ — X 
and. Equations (103) and (104), 

nx' 



X . d . 



f («2 - x'-'i' 



» = 2y -/a^ — x^ .dx = «^ (^2 ~ ^^^ 7 ~ ^' V"^ — ^''\ 

8 being the area of the entire segment. Denoting the chord AB 
by c, we have, 

whence, 



-v/a'^ - a;'^ = 



12 . s ' 



and we conclude, that tlie centre of gravity of a circular segment 
is on the radius drawn to the middle of the arc, and at a distance 
from the centre equal to the cube of the chord, divided by twelve 
times the area of the segment. 



108 



ELEMENTS OF ANALYTICAL MECHANICS. 



Replacing the value of s, and supposing x' to be zero, iii which 
case the segment becomes a semicircle, we shall find, 

c = 2 a. 



X, = 



4a 
3^* 



§ 122. — If the surface be one of revolution, about the axis x for 
instance, it will be symmetrical with respect to this axis ; hence, 

y, = 0; z^=0; 

and if F{xy) = 0, be the equation of a meridian section in the 
plane xy^ then will the area of an elementary zone comprised be- 
tween two planes perpendicular to the axis of revolution be. 



and therefore, Equations (92), 



2<7r 



Example 1. — Find 
the position of the 
centre of gravity of 
a right conical sur- 
face. 

The equation of 
the element in the 
plane xy, is, assum- 
mg the origin at the 
vertex. 



L"y''\J^ + dl-^-'^^ 


• . (105) 


s 


/"^•v/i + Z.-'^^ • • 


. . (106) 




y =z ax 



hence. 



2'Tr r^ax'^dx^l + a^ 
2'jf I ,, ax dx ^\ -{- a^ 



MECHANICS OF SOLIDS. 



109 



Example 2. — Required the posi- 
tion of the centre of gravity of 
a spherical zone. 

Assuming the origin at the 
centre, the equation of the me- 
ridian curve is, 



r?; 



whence, 



y dy =z — xdx, 
dy"^ x^ 




and, 



X, = 



/.■ 



x" 4- x' 



L 



ad X 



2 (a;" - x') 



Hence, the centre of gravity of a spherical zone, is at the middle 
point of a line joining the centres of its circular bases. And in the 
case of one base it is only necessary to make x'^ = a, which gives, 

x' -f a 

So that the centre of gravity of a zone of one base is at the middle 
of the ver-sine of its meridian curve. 

THE CENTRES OF GRAVITY OF VOLUMES. 



§123. — When it is the question to determine the centre of gravity 
of the volume of any body, we have 

dV z= d X . d y . d z, 
and Equations (92) become, 



II I II I II x.dy .dz .dx 

X J y J z 



110 ELEMENTS OF ANALYTICAL MECHANICS. 

n ,r nV-dy.dz.dx 

X J y J z 



y» = 



and, 



V 

/x' pyf pzf 
" I " J " ^'dy 'dz.dx 

-, = — ^ — '-V ' 

nxf pyf /»z' 

V = / „ ,, ,, dy .dz.dx. 

J X J y J z 



In which the triple integral must be extended to include the 
entire space embraced by the surface of the body ; this surface 
being given by its equation. 

If the volume be symmetrical with respect to any line, this line 
may be assumed as one of the co-ordinate axes, as that of ic ; in 
which case, if X represent the area of a section perpendicular to this 
axis, and x, its distance from the plane y z, then will Xdx^ be an 
elementary volume symmetrically disposed in regard to the axis ar, 
and Equations (92), become 

px' 

I ,, Xxd X 
^, =^^^-^ (107) 



= 0, 



and, 



z, = 0, 

V=fJnXdx (108) 



Example 1. — Find the position of the centre of gravity of a semi- 
ellipsoid, the equation of whose surface is 

The axes of the elliptical section parallel to the plane y z^ are, 



B 






MECHAXICS OF SOLIDS. Ill 

whence, 

X=.5C(1-J), 

and, Equations (107) and (108), 

= — A. 



f\BC,(l-^,)d, 



If the solid be one of revolution about the axis of a;, then, denoting 
by 

F{xy) = 0, (109) 

the equation of the meridian section by the plane x y, will 
and Equations (107) and (108), may be written, 



V 



I ,, 'It y"^ X dx 

~ — r^' ("0) 

f^,', *y'dx (HI) 



Example 1. — Required the position of the centre of r/ravity of a 
paraboloid of revolution. 

In this case, Equation (109), 

Fi.ry) zr: y^ -'2px^0, 

whence, 

V = tlir p I xd .-r, 



'irrp r x-^dsL 

r.^ ::::: t" 

2 -TT ;) j xdx 



= y"- 



112 ELEMENTS OF ANALYTICAL MECHANICS. 

Example 2. — Required the position of the centre of gravity of the 
volume of a spherical segment. 



whence, 



F{xy) = 2/2 _j_ ^2 _ ^2 _ 0^ 
V = ':r r\a^ - x^)dx 

J x" 

nx' 

I ^^ {a? — .r^) .x.dx 



It 
X, = 



or. 



'^ 1^ {a^ — x'^) d X 

^' ~ T L^3^^ - x"'^) -x' (3a2 - x'^)r 
and for a segment of one base, x'' = a, 

If the volume have a plane face, and be of such figure that the 
areas of all sections parallel to this face, are connected by any law 
of their distances fi'om it, the position of the centre of gravity, may 
also be found by the method of single integrals. 

Example 1. — Find the centre of gravity of any pyramid. 

Find by the method explained, the centre of gravity of the base 
of the pyramid, and join this point with the vertex by a straight line. 
All sections parallel to the base are similar to it, and will be pierced 
by this line in homologous points and therefore in their centres of 
gravity. Each section being supposed indefinitely thin, and its weight 
acting at its centre of gravity, the centre of gravity of the entire 
pyramid will, I^T, be found somewhere on the same line. 

Take the origin at the vertex, draw the axis x perpendicular to 
the plane of the base, and the plane xy through its centre of 



MECHANICS OF SOLIDS. 



113 



gravity; and let X represent anj section parallel to the base, then 
will Equations (92) become, 



X. = 






X dx 



y, = 



yr>x> 
,,Xy 



z, = 0, 



dx 




and, 



V = £!,Xdx, 
Represent by A the base of the pyramid, c its altitude, and let 



aar. 



be the equation of the line joining the vertex and centre of gravity 
of the base. 



Then, 



and for any frustum, 



A'.X'.'.c^'.x^, 
Ax^ 



X = 



rT- 



A r" 

-T. I /' x^ dx 



Ax^dx 



—, I " x'^dx 






Vi = 



a A p^' 

— ^Jz" ^^^ 



~f 



4 W'-^ - x'^y ' 



x^dx 



and for the entire pyramid, make x" = c, and x^ = 0, which give 

y, = T«c; 



114 ELEMEITTS OF ANALYTICAL MECHANICS. 

whence we conclude that the centre of gravity of a pyramid is on 
the line drawn from the vertex to the centre of gravity of the base, 
and at a distance from the vertex equal to threefourths of the length 
of this line. 

The same rule obviously applies to a cone, since the result is 
independent of the figure of the base. 

The weight of a body always acting at its centre of gravity, and 
in a vertical direction, it follows, that if the body be freely sus- 
pended in succession from any two of its points by a perfectly 
flexible thread, and the directions of this thread, when the body is 
in equilibrio, be produced, they will intersect at the centre of gravity ; 
and hence it will only be necessary, in any particular case, to deter- 
mine this point of intersection, to find, experimentally, the centre 
of gravity of a body. 

THE CENTKOBAETC METHOD. 



§124. — Eesuming the second of Equations (95) and (103), which 



are. 



ill which 



y/ 



^^-xA^ 



fn<^^\J 



1 4-'^*'' 



and 



in which 



y, = 



fUf''-y'')dx 



S = £!, [y" -y')dx', 
clearing the fractions and multiplying both menibers bj 2-^, \re 
shall have, 

2*.y,5 = ^',' 2^2/ ./d^^^^di\ . . . (112) 
2nty^s^ £1 'it{y"^ -y'^)dx .... (113) 



MECHANICS OF SOLIDS. 



115 



The second member of Equation (112) is the area of a surface 
generated by the revolution of a plane curve, ^Yh(>se extremities 
are given by the ordinates answering to the abscisses x' and x'\ 
about the axis x. In the first member, s is the entire length of 
this arc, and 2 cr y^ is the circumference generated by its centre of 
gravity. Hence, we have this simple rule for finding the area of a 
figure of revolution, viz. : 

Multiply the length of the generating curve hy the circumference 
described by its centre of gravity about the axis of rotation; the 
product will be the required surface. 

The second member of Equation (113) is the volume generated 
by a plane area, bounded by two branches of the same curve or 
by two different curves, and the ordinates answering to the abscisses 
x' and x", about the axis x. ■§, in the first member, is the generating 
area, and 2iry^ the circumference described by its centre of gravity. 
Hence, this rule for finding the volume of any figure of revolution, viz. : 

Multiply the generating area by the circumference described by its 
centre of gravity about the axis of rotation ; the product icill be the 
volume sought. 

Example \.— Required the measure of the surface of a right cone. 

Let the cone be generated by the 
rotation of the line A B about the , ^ 

line A C. The centre of gravity of 
the generatrix is at its middle point 
G. and therefore, the radius of the 
circle described by it will be one- 
half of the radius (7i>, of the circu- 
lar base of the cone. Hence, 

BC.AB 



c 



y, . s = 2 ^ . 



rr BC.AB. 



Example 2. — Find the volume of the cone. 

The area of the generatrix ABC, is ^ B C . A C \ and the radius 
of the circle described by its centre of gravity is -J- B C. Hence, 



2^y,s =: ^itBC 



BC.AC 



BC^.AC 



116 



ELEMENTS OF ANALYTICAL MECHANICS. 



CENTRE OF INEKTIA. 

§ 125. — When the elementary masses of a body exert their forces 
of inertia simultaneously and in parallel directions, they must expe- 
rience equal accelerations or retardations in the same time, and the 
factor 

(Ps 

in the measures of these forces, as given in Equation (13), must be 
the same for all. Substituting these measures for P', P", dzc, in 
Equations (62), we find, 



d^s 



' 2 7n X 



X =z 



dt'' 




Smx' 


dh 

dt:^ 


.2m 


~ 2m ' 


dh 
dt'' 


'^my' 


I.my\ 
~ 2m ' 


dh 
dfi 


• Sm 


dh 

dfi 


'1ms' 


2 ms' 


dh 
di^ 


.2m 


2 m 



(114) 



Whence, Equations (88), the centre of inertia coincides with the 
centre of gravity when the latter force is constant, both being at the 
centre of mass. In strictness, however, the centre of gravity is 
always below the centre of inertia ; for when the variation in the 
force of gravity, arising from change of distance, is taken into 
account, the lower of two equal masses will be found the heavier. 
And in bodies whose linear dimensions bear some appreciable propor- 
tion to their distances from the centre of attraction, the distance 
between these centres becomes sensible, and gives rise to some curious 
phenomena. 



MECHANICS OF SOLIDS, 



117 



MOTION OF THE CENTEE OF INEETIA. 



§ 126. — Substitute in Equations (^), the values of d"^ x, d^y^ and d'^z, 
given by Equations (34), and we have, because dt is constant, and 
d'^x^^ d'^y^ and d'^z^^ will each be a common factor for all the elemen- 
tary masses, 



d'^x 
2F cos ct — M- 

HFcos (3 — M- 

^P cos y-M.-^ 



dfi 


dt^ 


d^y, 
dt^ 


1 

dt^ 


d^z, 


1 



di"^ 



Hm.d^x' =z 0, 
. 2 m . J2 y' — 0, 
2 m . (f2 2' - 0. 



m which M, denotes the entire mass of the body, being equal to 2 m. 
Denote by x, y, 2, the co-ordinates of the centre of inertia referred 
to the movable origin, then. Equations (114), 

M. X = ^mx\ 
M.y =z ^my\ 
M. z = '^m z\ 



and differentiating twice. 



M. d'^x rr 2 m . d-^x', ^ 
M.d?y = 2 w . dhj\ 
M.dh — ^m,dH\ 



which substituted in the preceding Equations, give. 



2 p. cos a -Jf.^ -M''^ = 0, 

dfl c/r 

2P.cos,/3-if.^ -M-%=(), 

dl^ dp 

2P.cos7-if. -^ - M---l~ ^. 0, 

^ dt^ dl^ ' 



(115) 



(116) 



118 ELEMEITTS OF ANALYTICAL MECHAISTICS. 

and if the movable origin be taken at the centre of inertia, then 
will, 

dhc = 0, dfy = 0, dfz-^\ 

and Xj^ y^ , z,^ will become the co-ordinates of the centre of inertia 
referred to the fixed origin, and we have, 

2 P. cos a - M'^ = 0, 



2P.COS/3 _ if. ^ z:. 0. 

2P.cosy-J!f.^ = 0;J 



(117) 



Equations which are wholly independent of the relative positions 
of the elementary masses m\ m" &c., since their co-ordinates x\ y\ 
z\ &;c., do not enter. It will also be observed that the resistance of 
inertia is the same as that of an equal mass concentrated at the 
body's centre of inertia. 

Whence we conclude, that when a body is subjected to the action 
of any system of extraneous forces, the motion of its centre of inertia 
will be the same as though the entire mass were concentrated into 
that point, and the forces applied without change of intensity and 
parallel to their primitive directions, directly to it. 

This is an important fact, and shows that in discussing the motion 
of translation of bodies, we may confine our attention to the motion 
of their centres of inertia regarded as material points. 



ROTATION ABOUND THE CENTRE OF INERTIA. 

g 127. — Now, retaining the movable o'-'n-in at the centre of inertia, 
substitute in Equations (i)), the values of d'^x, d'^ij^ and d^z^ as given 
by Equations (34), and reduce by the relations, 



M .y ^ ^ m . 7j' =:0, 
M.'z = :^m.z' =0\ 



MECHANICS OF SOLIDS. 119 

and we have, 

SP.icosy.y' - 0OSI3.Z') -:Em. {—■/ -~^-z' ) =0; 



(118) 



from whicli all traces of the position of the centre of inertia have 
disappeared, and from which we infer that when a free body is acted 
upon by any system of forces, the body will rotate about its centre 
of inertia exactly the same whether that centre be at rest or in 
motion. 

g 128. — And we are to conclude. Equations (117) and (118), that 
when a body is subjected to the action of one or more forces, it will 
in general, take up two motions — one of translation, and one of rota- 
tion, each being perfectly independent of the other. 

§129. — Multiply the first of Equations (117), by ?/^ , the second by 
x^ , and subtract the first product from the second ; also, the first by 
z , the third by x^ , and subtract the second of these products from 
the first ; also the third by y^ , and the second by z^ , and subtract 
the second of these products from the first, and we have, 



/f/^7/ d^x \ 

2{Fcosi3).x-^{Pco,a).,j-3f-{jj^-x,--^^,j^) =0, 
2 (P cos a). 2, -2 (P cos y).^, -3/. (-^-2, - -^-xj =0, 
2(Pcos7).y,-2(Pcos/3).2,-3/. (^.y,-^.^,) =0; 



(119) 



Equations from which may be found the circumstances of motion 
of the centre of inertia about the fixed origin. 



120 ELEMENTS OF ANALYTICAL MECHANICS. 



MOTION OF TEAXSLATION. 



§ 130. — Regarding the forces as applied directly to the centre of 
inertia, replace in Equations (117), the values 2 P . cos a, IF. cos (3^ 
and 2P.COS7, by X, F, and Z, respectively, and we may write, 



(120) 



from which the accents are omitted, and in which x, y, and 0, must 
be understood as appertaining to the centre of inertia. 



-— s=».^ 




''— s = »^ 




'-^■^-o-- 





GEJSTEEAI. THEOEEM OF WORK, VELOCITY AND LIYIXG- FOECE. 

§131. — Multiply the first of Equations (120) by 2dx, the second 
by 2c?y, the third by 2dz, add and integrate, we have 



2f{Xdx 4- Tdy + Zdz) - M, 
But, 



dx"^ + di/^ + dz^ 
dJ^ 



■\-C=0. 



dx"^ 4- dy"^ + dz"^ ds^ 



df^ 



df^ 



Y2, 



whence, 



2f(Xdx 4- Tdy + Zdz) - M.V^-i- C = 



(121) 



The first term is, § 101, twice the quantity of work of the ex- 
traneous forces, the second is twice the quantity of work of the 
inertia, measured by the living force, and the third is the constant 
of integration. 

If the forces X, Y, Z, be variable, they must be expressed in 
functions of x, y, z, before the integration can be perfoiraed« 



MECHANICS OF SOLIDS. 121 

Supposing this latter condition fulfilled, and that the forms of the 
functions are such as make the integration possible, we may write, 

F{xyz)- -i-i/.F2 + C"= 0, .... (122) 

.nd between the limits x^ y, z, and x^ y/ z/ , 

F{x; y/ z/) - F (x,y,z) = ^M {V^ -V^^) . . (123) 

whence we conclude, that the quantity of work expended by the 
extraneous forces impressed upon a body during its passage from one 
position to another, is equal to half the difference of the living forces 
of the body at these two positions. 

We also see, from Equation (123), that whenever the body 
returns to any position it may have occupied before, its velocity will 
be the same as it was previously at that place. Also, that the 
velocity, at any point, is wholly independent of the path described. 

§ 132.— If 

Xdx + Tdy + Zdz = 0, 
the extraneous forces will, §101, be in equilibrio, and 

"2777' 



../ 



M 



that is, the velocity will be constant, and the motion, therefore, 
uniform. 



CENTRAL FORCES. 

§133. — Forces which act towards a given point, either at rest or 
in motion, and the intensities of which depend upon the distance 
from that point, arc called central forces. The forces of nature are 
of this description. 

It will always be possible to find the velocity,— that is, to integrate 
the first term in Equation (121), when the extraneous forces are 
directed to fixed centres, and their intensities are expressed in fimctions 
of the body's distances from these centres. 



122 ELEMENTS OF ANALYTICAL MECHANICS. 



For, denote the constant 
co-ordinates of the fixed cen- 
tres by a b c, a' h' c', &c., 
and the distances from the 
body to these centres by 
_p, p\ &c., then ^Yill 




X — a ^ y — h z — c 
cos a = , cos p = ? cos y = > 



and the same for the other centres, whence, 



x = p.^^:^ + p'..^^ + &c. 
p p 

P P' 



P P 



Multiplying the first by dx^ the second by dy^ the third by dz, 
adding and integrating, there will result, 



fp. (^,,+t±,,+^,,) 

*J \ 2^ p p / 

L + &c; 
but. 



whence. 



p = Vi^ - «)' + (2/ - b)^ + (^ - c)2, 



X — a y — h z — c 

dp — dx -\- • . dy -\ dz ; 

P P P 



MECHA^^ICS OF SOLIDS. 123 

and the same for dv' ; Avhich substituted iii the precedmg equation 
for the work, gives, 

/ {Xdx + Ydy + Zdz) = f {Pdp + P'dp' + &c.) ; 

but, by hypothesis P is a function of }) : also, P' of p\ &c. ; each 
term is, therefore, a function of a single variable ; whence, the truth 
of the proposition. 

Substituting in Equation (1-1), we get, 

f [Pdp + P'dp' + &c.) - ^MV J^ C'=0 . . (123)' 



STABLE AXD UNSTABLE EQUILLBKniM. 

§ 134. — Resuming Equation (123), omitting the subscript accents, 
and bearing in mind that the co-ordinates refer to the centre of 
inertia, into which we may suppose for simplification the body to be 
concentrated, we may write, 

■l-iV/F'2 - ^MV = F{x'y'z') - F{xyz), 

in which 

F{xyz) = /{Xdx + Ydy + Zdz), 

and 

dF{xyz) = Xdx + Ydy -f Zdz, 

Now, if the limits x' y' z' and xyz be taken very near to each 
other, then will 

x' — X ^ dx', y' = y + dy) z' = z -^ dz; 

which substituted above, give 

I i/F'2 - IMV = F(x + dx, y -{- dy, z -^ dz) - F{xyz), 

and developing by Taylor's theorem, 

( Adx -f Bdy -{- Cdz 
^ ^ \^ A' dx"" -\- B' dy-^ + &c. 4- i>, 

in which D denotes the sum of the terms involving the higher 
powers of dx, dy and dz. 



124: ELEMENTS OF ANALYTICAL MECHANICS. 

If -J J/ F^ be a maximum or minimum, then will 

Adx + Bdij -\- Cdz =zO', (123)" 

and since 

Adx + Bdy + Cdz = dF{xyz) = Xdx + Ydy + Zdz, 
we have, 

Xdx + Ydy + Zdz = 0. 

But w^hen this condition is fulfilled, the forces will. Equation (69), 
be in equilibrio ; and we therefore conclude that whenever a body 
whose centre of inertia is acted upon by force c; not in equilibrio, 
reaches a position in which the living force or the quantity of 
work is a maximum or minimum, these forces will be in equilibrio. 

And, reciprocally, it may be said, in general, that when the forces 
are in equilibrio, the body has a position such that the quantity of 
action will be a maximum or minimum, though this is not always 
true, since the function is not necessarily either a maximum or a 
minimum when its first differential co-efficient is zero. 

§ 135. — Equation (123)", being satisfied, we have 

^if F'2 _ i-if F2 =. ± {A'dx^ + B'dy^ + &c. + i)) . • • (124) 

The upper sign answers to the case of a minimum, and the lower 
to a maximum. 

Now^, if V be very small, and at the same time a maximum, V 
must also be very small and less than F, in order that the second 
member may be negative ; whence it appears that whenever the system 
arrives at a position in which the living force or quantity of work is 
a maximum and the system in a state bordering on rest, it cannot 
depart far from this position if subjected alone to the forces which 
brought it there. This position, which we have seen is one of equi- 
librium, is called a position of stable equilibrium. In fact, the quantity 
of work immediately succeeding the position in question becoming 
negative, shows that the projection of the virtual velocity is negative, 
and therefore that it is described in opposition to the resultant of the 
forces, which, as soon as it overcomes the living force already existing, 
will cause the body to retrace its course. 



MECHANICS OF SOLIDS. 125 

1 136, — If^ on the contrary, the body reach a position in %Yhich the 
quantity of work is a minimum, the upper sign in Equation (124), 
must be taken, the second member will always be positive and there 
will be no limit to the increase of V\ The body may therefore 
depart further and further from this position, however small V may be ; 
and hence, this is called a position of unstable equilibrium. 

R 137. — If the entire second member of Equation (124), be zero, 
then will, 

■l-jlf F'2 - iif F2 = 0, 

and there will be neither increase nor diminution of quantity of work, 
and whatever position the body occupies the forces will be in equili- 
brio. This is called equilibrium of indifference. 

g 138. — If the system consist of the union of several bodies acted 
upon only by the force of gravity, the forces become the weights 
of the bodies which, being proportional to their masses, will be con- 
stant. Denoting these weights by W'^ W'\ W"\ &c., and assum- 
ing the axis of z vertical, we have from Equations (87), 

Rz^ :=^ W'z' + W"z" + W"'z"' + &c., 

in which i?, is the weight of the entire system, and z^ the co-ordi- 
nate of its centre of gravity; and differentiating, 

Rdz, = W'dz' + W"dz" + W"'dz'" -f &c. . . . (125) 

Now, if z, be a maximum or minimum, then will 

W dz' + W" dz" -h W'dz'" -f &c. = 0, 

which is the condition of equilibrium of the weights. Whence, we 
conclude that when the centre of gravity of the system is at the 
highest or lowest point, the system will be in equilibrio. 

In order that the virtual moment of a weight may be positive, 
vertical distances, when estimated downwards, must be regarded as 
positive. This will make the second differential of z^ , positive at 
the limit of the highest, and negative at the limit of the lowest 
point. Tlie equilibrium will, therefore, be stable when the centre of 
gravity is at the lowest, and unstable when at the highest point. 



126 ELEME^^TS OF ANALYTICAL MECHANICS 

Integrating Equation (125), between the limits z^ = H^ and 
z^ = H', z' = h, and z' = h\ &c., and we find, 

R{II, - H') = W'{h, - h') + W" (A,, - h") + &c. ; . (126) 

from which we see that the work of the entire weight of the system, 
acting at its centre of gravity, is equal to the sum of 'the quantities 
of work of the component weights, which descend diminished by the 
sum of the quantities of work of those w^hich ascend. 



INITIAL CONDITIONS, DIRECT AND EEVEKSE PROBLEM. 

§ 139. — By integrating each of Equations (120) twice, we obtam 
three equations involving four variables, viz. : x^ y^ z and t. By 
eliminating i, there w^ill result two equations between the variables 
iT, y and 0, which will be the equations of the path described by 
the centre of inertia of the body. 

§ 140. — In the course of integration, six arbitrary constants will 
be introduced, whose values are determined by the initial circum- 
stances of the motion. By the term initial^ is meant the epoch 
from which t is estimated. 

The initial elements are, 1st. The three co-ordinates which give 
the position of the centre of inertia at the epoch ; and 2d. The 
component velocities in the direction of the three axes at the same 
instant. 

The general integrals determine the nature only, and not the 
dimensions of the path. 

§141. — Now two distinct propositions may arise. Either it may 
be required to find the path from given initial conditions, or to 
find the initial conditions necessary to describe a given path. 

In the first case, by differentiating the three integrals with respect 

. . . , . dx dy dz 

to ^, we obtam three equations mvolvnig rr, y, 2, ■—-> —5 —■> t, 

do Qj Z Q/ V 

and the arbitrary constants ; making t equal to zero, and giving 
the initial elements their values, there will result three more equa- 



MECHANICS OF SOLIDS. 



127 



tioiis involving the arbitrary constants and known quantities. From 

these six equations we may find the arbitrary constants, and the 

problem is completely solved. 

In the second case, vre shall have given two equations involving 

d'^ X cP 7/ d'^z 
X, y, 0, and these also contain -r^? -j^^ -y-^' or X, Z, Z, which 

shows that the problem is indeterminate. 

But Equation (121) being differentiated and divided by the dif- 
ferential of one of the variables, say d x, gives 



M/^^^x+r/'-' 



which is a third equation involving X, Y, Z, and V. By assuming 
a value for any one of these four quantities, or any condition con- 
necting them, the other three may be found in terms of x, y and z. 



VEETICAL MOTIOX OF HEAVY BODIES. 

§ 142. — When a body is abandoned to itself, it falls toward the 
earth's surface. To find the circumstances of naotion, resume Equa- 
tions (120), in which the only force acting, neglecting the resistance 
of the air, will be the weight =. Mg ; and we shall have, Equa- 
tions (in), 

2 P cos a = X = Mg . cos a ; 
2 P cos /3 r= F = Mg . cos Q ; 
2 P cos 7 = Z = Mg . cos 7 ; 

in which M denotes the mass of 
the body. The force of gravity 
varies inversely as the square of 
the distance from the centre of 
the earth, but within moderate 
limits may be considered invaria- 
ble. The weight will therefore be 
constant during the fall. 

Take the co-ordinate z vertical, 
and positive when estimated downwards, then will 

cos a = ; cos 1^ =1 \ cos v :^ 1 , 



128 ELEMENTS OF AJ^TALYTICAL MECHANICS. 

and Equations (120) become, after omitting the common factor M^ 



and integrating, 



cPx cf^y d^z 

77^' Tt^"^^' Tfi"^^' 



dx dy 

dt ^' dt y' 



p^=v =:gt-hu^ (128) 

in which v is the actual velocity in a vertical direction. 
Making ^ = 0, we have 

dz _ 
dt~^''' 

The constants u , u and u , are the initial velocities in the 
directions of the axes x, y and 0, respectively. Supposing the first 
two zero, and omitting the subscript 0, from the third, we have, 

dx ^ dy 

v=^^ = fft + u (129) 

Integrating again, we find 

X = C; y = C, 
z = lfft^ + ut+ C", 

and if, when i =: 0, the body be on the axis 2, and at a distance 
below the origin equal to a, then will 

.r r= ; 2/ = ; 
z = \gt'^ J^ ut ^ a ' .. . . . (130) 

If the body had been moving upwards at the epoch, then would 
u have been negative, and, Equations (129) and (130), 

V — gt — u (131) 

z = \9f - ut + a ' . .. . . (132) 



MECHANICS OF SOLIDS. 129 

If the body had moved from rest at the epoch and from the 
origm of co-ordinates, then would v be the actual velocity generated 
by the body's weight, and z = A, the actual space described in the 
time t J and Equations (129) and (130) would become, 

V = gt (133) 

h =igtK (134) 

and eliminating i, 

V = y^Wih (135) 

whence, we see that the velocity varies as the time in which it is 
generated ; that the height fallen through varies as the square of the 
time of fall; and that the velocity varies directly as the square root 
of the height. 

The value of k, is called the height due to the velocity v ; and 
the value v, is called the velocity due to the height h. 

If, in Equation (132), we suppose a = 0, we shall have the case 
of a body thrown vertically upwards with a velocity u, from the 
origin, and we may write, 

V = fft — ^{, . (136) 

z^igP -ui] (137) 

when the body has reached its highest point, v will be zero, and we 
find, 

gt — u = 0; 
or. 



u 

/ = — ; 

which is the time of ascent; and this value of i, in Equation (137), 
will give the greatest height, h = z, to which the body will attain, 

'' = -i •• (^■^«) 

§ 143. — In the preceding discu'ssion, no account is taken of the 
atmospheric resistance. For the same body, this resistance varies as 

9 



130 ELEMENTS OF Al^ALYTICAL MECHANICS. 

the square of the velocity, so that if ^, denote the velocity when the 
resistance becomes equal to the body's weight, then will 

M . g .v"^ 

be the resistance when the velocity is v, and in Equations (117), we 
shall have, 

2 P COS OL =z X = M g cos a + ^g • t^ • cos a', 

^2 

2Pcos/3= F= J[f^ cos/3 + J[f^ .—. cos ^', 

^2 

2 P cos y :=z Z = M g cos y + Jlfy • — • cos y' ; 

taking the co-ordinate ^, vertical and positive downward, then will, 

cos a = cos oJ z=L 0, 
cos /3 = cos /3' = 0, 
cos y =. \^ cos 7' = — 1 ; 

and, supposing the body to move from rest, Equations (120), give, 



d z civ 

Omitting the common factor M^ and replacing -—^ by its value — , 

whence, 

¥-.dv h /' dv , dv \ ,,«,^^ 

Integrating and supposing the initial velocity zero, 

fft = ik.\og i±^ (140) 



MECHANICS OF SOLIDS. 131 

which gives the time in terms of the velocity; or reciprocally, 



k -\- V k 



h — V 



1g t 

(141) 



in which e^ is the base of the Naperian system of logarithms, and 
from which we find. 



/ 9_t _sj\ 
^^^^-^"-iA, (142) 



s = --log^ (e"* + r *^ (143) 



9_t 

which gives the velocity in terms of the time. Substituting for v, 
its value -^? integrating and supposing the initial space zero, we 

have 

J^2 - ^ « St 

9 

Multiplying Equation (139) by 

dz 

Tt='' 

we have, 

, W-.v ,dv 

and integrating, observing the initia} conditions as above, 

which gives the relation between the space and velocity. 

_ 2l 
As the time increases, the quantity e * becomes less and less, 

and the velocity, Equation (142), becomes more nearly uniform ; 

for, if t be infinite, then will 

_ LL 
e * = 0, 

and, Equation (142), 

V =: A; ; 

making the resistance of the air equal to the body's weight. 



132 ELEMENTS OF ANALYTICAL MECHANICS. 

§144. — If the body had been mov-iiig upwards with a velocity 
V, then, taking z positive upwards, would, Equations (120), 

civ d"^ z 

substituting -— for -r-y' and omitting the common factor, we find, 



Tc .dv gdt 



F + ^2 1c ' 

integrating, 

and supposing the initial velocity equal to a, we find 



C = tan — ? 

rC 



(145) 



and, 



Taking the tangent of both members and reducing, we find 

a — tc . tan -— 
^ = i.. 1 (147) 

at 
k + a. tan — 

K 

which may be put under the form, 

gt T ' g^ 

a . cos ^; ^ . sm — 

^^jc -^ "- .... (148) 

. Q i ■. gi 
a . sm ^ + rt^ . cos — 

Substituting for v its value — ? integrating, and supposing the 
initial space zero, we have 

g 



2 = — . log 



(t-'"t + -I)- • • •(^^^> 



MECHANICS OF SOLIDS. 133 

Multiplying Equation (145), by 

dz 

and we have, 

Ic^ .V .dv 



g .dz = 



k^ 



and integrating, with the same initial conditions of v being equal to 
a, when z is zero, there will result, 

"ii--^iu <■=«) 

§ 145. — If we denote by h, the greatest height to which the body 
will ascend, we have z = h, when v ^ 0, and hence, 

Finding the value of ^, from Equation (146), we have, 

' = 7('-"t-'-"t) • • ■ • (i^^> 

from which, by making t; = 0, we have, 

«. = 7 • ton -J (153) 

which is the time required for the body to attain the greatest eleva- 
tion. Having attained the greatest height, the body will descend, and 
the circumstances of the fall will be given by the Equations of §143. 
Denoting by a', the velocity when the body returns to the point of 
starting, Equation (144), gives, 

• loj 



2^7 * F _ ,^n 

and placing this value of h equal to that given by Equation (151), 
there will result, 

^-2 F + a2 



F - a'2 F 



134 ELEMENTS OF ANALYTICAL MECHANICS. 



whence, 



,'2 — n2 ^! ; 



a? + A;2 

that is, the velocity of the body when it returns to the point of 
departure is less than that with which it set out. 
Making v = a' m Equation (140), we have, 

Jc , h -{- a' 
and, substituting for a\ its value above. 



t = —— . log ■ ^ , • • • • (154) 

a value very different from that of ^^, given by Equation (153), for 
the ascent. 

Multiplying both numerator and denominator of the quantity whose 
logarithm is taken, by -y/ a^ -f k'^ — a, the above becomes, 

t =^.log— ^ (155) 

^ ^ -x/k^ + a2 -a 

Adding Equations (153) and (155), we have, 

^ r -1 a , , k -I 

^ + if - = — tan — + log ~^==== 

• 
or, making t = t^ -^ t ^ 

^ = tan"' 4- + log — — ^- . • . (156) 

If a ball be thrown vertically upwards, and the time of its 
absence from the surface of the earth be carefully noted, t will be 
known, and the value of k may be found from this equation. This 
experiment being repeated with balls of different diameters, and the 
resulting values of k calculated, the resistance of the air, for any 
given velocity, will be known. 



MECHANICS OF SOLIDS. 



135 



PKOJECTILES. 

§ 146. — Any body projected or impelled forward, is called a ;pro- 
jectile^ and the curve described by its centre of inertia, is called a 
trajectory. The projectiles of artillery, which are usually thrown with 
great velocity, will be here discussed. 

§ 147. — And first, let us consider what the trajectory would be 
in the absence of the atmosphere. In this case, the only force which 
acts upon the projectile after it leaves the cannon, is its own weight ; 
and, Equations (117), 

2 P cos oL = X = Mg cos a, 
2 P cos /3 = Y z:^ Mg cos /3, 
2 P cos 7 = Z — Mg cos /. 



Assuming the origin 
at the point of de- 
parture, or the mouth 
of the piece, and 
taking the axis z 
vertical, and posi- 
tive upwards, then 
will 



cos a =: ; cos /3 = ; cos 7 




if. 






0; >/.S = 0; M.^^, 



1 ; and. Equations (120), 
d-^z 



Mg-, 



and integrating, omitting if, 



dy 



dz 



d t 



at =%;.77 = -^' + »" 



(157) 



Integrating again, and recollecting that the initial spaces are zero, we 
have, 



x = u^'t', y-U't', z = ~^gi2 -\. u^* ( 



(158) 



136 ELEMENTS OF ANALYTICAL MECHANICS, 

and eliminating t^ from the first two, we obtain, 

u 

X 

which is the equation of a right line, and from which we see that 
the trajeotory is a plane curve, and that its plane is vertical. 

Assume the plane zx^ in this plane, then will y = 0, and Equa- 
tions (158), become, 

X t=z u^' i\ z =z — igt^ -}- u^' t. . . . (159) 

Denote by F, the velocity with which the ball leaves the piece, 
that is, the initial velocity, and by a, the angle which the axis of the 
piece makes with the axis x, then will, 

V. cos a, and V . sin a, 

be the lengths of the paths described in a unit of time, in the direc- 
tion of the axes x and z, respectively, in virtue of the velocity V ; 
they are, therefore, the initial velocities in the directions of these 
axes; and we have, 

M = Fcosa: u == F.sina; 
which, in Equations (159), give 

X —V. 0,0^0.. t\ z =-^gi^ '\- V .%\\\o.,t . . (160) 

and eliminating ^, we find 

o X 
z =z X tan a — 



2 F2 . cos2 a ' 
or substituting for F its value in Equation (135), 



z = X tan a — — — ^ -— (161) 

4 h . cos^ a ^ ' 



which is the equation of a parabola. 



MECHANICS OF SOLIDS, 



137 



§ 148. — The angle a is 
called the angle of projec- 
tion ; and the horizontal 
distance A D, from the 
place of departure A^ to 
the point i>, at which the 
projectile attains the same 
level, is called the range. 

To find the range, make z = 0, and Equation (161) gives 

a: = 0, and a; = 4 /i sin a cos a = 2 A sin 2 a, 

and denoting the range by i?, 

i2 = 2 7i . sin 2 a 




(162) 



the value of which becomes the greatest possible when the angle 
of projection is 45°. Making a = 45°, we have 

R = 2h (168) 

that is, the maximum range is equal to twice the height due to 
the velocity of projection. 

From the expression for its value, we also see that the same 
range will result from two different angles of projection, one of which 
is the complement of the other. 

§ 149. — Denoting by v the velocity at the end of any time ^, we 
have, 

'^'^ ^ m? ~ If 

or, replacing the values o^ dz and dx^ obtained from Equations (160), 
^2 _ 72 _ 2 F.y.^f.sina + 5r2/2 .... (164) 

and eliminating ^, by means of the first of Equations (160), and 
replacing F^, in the last term by its value 2^ A, 



y2 —. Y2 _ 2g . tan a . x -\- g 



2 h . cos^ a 



(105) 



138 ELEMENTS OF ANALYTICAL MECHANICS. 

in which, if we make x = 4 A . sin a cos a, we have the velocity at 
the point D, 

which shows that the velocity at the furthest extremity of the range 
is equal to the initial velocity. 

Differentiating Equation (161), we get 

-^ = tan ^ = tan a ~ ^ . ^ ^ .... (166) 
ax 2h.cos^a, ^ ' 

in which ^ is the angle which the direction of the motion at any 
instant makes with the axis x. 
Making tan ^ =: 0, we find 

a; = 2 A . cos a . sin a, 

which, in Equation (161), gives 

z = h. sin^ a, 

the elevation of the highest point. 

Substituting for x, the range, 4 h cos a sin a, in Equation (166), 

tan ^ r= — tan a, 

which shows that the angle of fall is equal to minus the angle of 
projection. 

g 150. — The initial velocity V being given, let it be required to 
find the angle of projection which will cause the trajectory to pass 
through a given point whose co-ordinates are x = a and z =: b. 

Substituting these in Equation (161), we have 



b =z a tan a - 

from which to determine a. 
Making tan a = 9, we find 

cos'^ a = 



4 h . cos^ a 



1 + 9' 



MECHANICS OF SOLIDS, 
which in the equation above, gives 



139 



whence, 



2Ji \ 

Q) = tan <x = — zb - ■\/4: h'^ — Ahb — a? 
^ a a 



(167) 



The double sign shows that the object is attained by two angles, 
and the radical shows that the solution of the problem will be 
possible as long as 

4A2>4A6 + a2. 

Making, 

4 ^2 _ 4 ;^\ 5 _ ^2 ^ Q^ 

the question may be solved with only a single angle of projection. 
But the above equation is that of a parabola whose co-ordinates are 
a and 6, and this curve being con- 
structed and revolved about its vertical 
axis, will enclose the entire space 
within which the given point must be 
situated in order that it may be struck 
with the given initial velocity. This 
parabola will pass through the farthest 
extremity of the maximum range, and 
at a height above the piece equal to h. 




JL 



J) 



§ 151. — Thus we see that the theory of the motion of projectiles 
is a very simple matter as long as the motion takes place in vacuo. 
But in practice this is never the case, and where the velocity is con- 
siderable, the atmospheric resistance changes the nature of the tra- 
jectory, and gives to the subject no little complexity. 

Denote, as before, the velocity of the projectile when the atmos- 
pheric resistance equals its weight, by k^ and assuming that the 
resistance varies as the square of the velocity, the actual resistance 
at any instant when the velocity is v, will be, 



M .g .v^ 



= Mcv^ 



14:0 ELEMEI^TS OF ANALYTICAL MECHANICS, 

by making, 

The forces acting upon the projectile after it leaves the piece 
being its weight and the atniospheric resistance, Equations (120), 
become, 

M • -r-ir = ^9 - cos a + Mc . v^ . cos a! 
a V' 

cP If 

M' -TY = ^9 . cos /3 + Mc . v^ , cos /3', 

d? z 
M' -—^ ^=1 M9 .Q^Q^y -\- Mc.v^ .do^y'. 

Taking the co-ordinates z vertical, and positive when estimated 
upwards, 

cos a r= ; cos /3 = ; cos y =■ — 1, 

and because the resistance takes place in the direction of the trajec- 
tory, and in opposition to the motion, if the projectile be thrown in 
the first angle, the angles a', /3', and 7', will be obtuse, 

dx dy dz 

cos a' = — : cos p = r— ; cos 7 == — , 

ds ds * d s 

and the equations of motion become, after omitting the common 
factor M, 



di^ 


z= 


— € ' 


.v^. 


dx 
ds 


y 


d^y 

dt'^ 


=: 


— C 


.2;2. 


dy 
ds 


J 


d-'z 
dt^ 


= 


-9 


— c 


.v^. 


dz 
ds 



From the first two we have, by division, 

d^y d^x 
dy ~ dx ' 



MECHANICS OF SOLIDS. 141 

and bj integration, 

log c?y = log dx -{- log C'y 

and, passing to the quantities, 

dij = Cd X. 

Integrating again, we have, 

y=Cx-\-C'', 

in which, if the projectile be thrown from the origin, (7' = 0, thus 
giving an equation of a right line through the origin. Whence we 
see that the trajectory is a plane curve, and that its plane is vertical 
through the point of departure. 

Assuming the plane z rr, to coincide with that of the trajectory, 
and replacing v"^^ by its value from the relation, 



df- 



we have, 



~d~fi 
d'^z 



— c • 



d 8 d X 
~dt"dJ 



ds dz 



(168) 



From the first we have, 



df- ds 

= -^'77' 



d X 
~dT 



and by integration. 



loi 



dx 
~dt 



=. - c.s -\- C, 



Denoting by e, the base of the Naperian system of logarithms, 
and making C = log yl, the above may be written, 

log -^ = - c . 5 X log e -f log A, 



142 ELEMENTS OF ANALYTICAL MECHANICS. 

and passing from logarithms to the quantities, 

// r — c s 

S-=^- (^«^) 

Denoting by V, the initial velocity, and by a, the angle of pro- 
jection, we have, by making s = 0, 

~ — = A z= V cos a, 
a ^ 

which substituted above, gives 

dx ""''* 

— = F.cosa.e (170) 

To integrate the second of Equations (168), make 

Tt = ^-Tt' (^^^) 

in which p is an additional unknown quantity. 

Differentiating this equation, dividing by dt, and eliminating 

from the result, -ry' by its value in the first of equations (168), 
we have. 



d^z dp dx ds dx 

d¥ ~ dTt ' dTt ~ ^ '^ ' ITt'Ti 



and substituting this value in the second of Equations (168), we 
d^ 
Tt 



dz . 
have, after eliminating — by its value, obtained from Equation (171), 



%-%--^ (-) 



and dividing this by the square of Equation (170), 



dp 

d t a 2 c» 

^ ^ - „/ , • e (173) 

dx K 2 cos^ a ^ ' 

Tt 



MECHAITICS OF SOLIDS. 143 

but regarding z and 'p as functions of x^ we have. Equation (171), 



d 



dtdz 

dt 

and, 

d]p 

d t dp 
dx ~ dx 
^dT 

whence, making V^ = 2ffhy Equation (173) becomes 



dp 



2CS 

5 



d X 2h. cos^ a 

and multiplying this by the identical equation 



(175) 



d X . yT~+^ = ds, 
obtained from Equation (174), we find, 



y 1 -]- p'^.dp = 



2cs 

e • ds 



2 h . cos^ a ' 
and integrating, 



2es 

e 



p. V1+// + log {p + ^r^^) = C - ^^i^^^^,^ ; (176) 

in which C is the constant of integration ; to determine which, make 
3 = 0', this gives p — tan a ; and 



^ "" 2 c A c os^ a + ^""'^ ^ Vl + tan^ a + log (tan a + VT + tan^a) . (177) 
From Equation (175) we have, 



■2 c s 



dx = ^ 2A.cos2a.e • dp ; 
from Equation (171), 

dz = p .dx 'j 



144: ELEMENTS OF ANALYTICAL MECHANICS. 

from Equation (n2), 

g df" =. — dx .dp\ 

and eliminating the exponential factor by means of Equation (176), 
we find, 

c.dx = ^ ^ ^ ; . (178) 



c,dz = —_ '^^ ;. (179) 



i"/! 


+ P' 


+ log (p + V^+ p^) - 

pdp 


- G 


P^i 


+ j>2 


+ log(i> + -/l +y)- 
-dp 


- 



V^.dt^ ' ■ ^ =^; . (180) 

Jg-p vTT7^ - log {p + -/TTF) 

Of the double sign due to the radical of the last equation, the 
negative is taken because ^, which is the tangent of the angle made 
by any element of the curve with the axis of a?, is a decreasing 
function of the time t. 

These equations cannot be integrated under a finite form. But 
the trajectory may be constructed by means of auxiliary curves of 
which (178) and (179) are the differential equations. Erom the first, 
we have, 

dx = T .dp-, . (181) 

and from the second, 

dz = T.p.dp', (182) 

in which, 

T^ — ^ ^ ;. (183) 

' P'y/l+p' ^\og{p-\- ^l+p-') - C 

and dividing Equations (181) and (182), by dp, 

-^ = ^'- • • (^^^) 



MECHANICS OF SOLIDS. 



145 



Now, regarding rr, ^, and 2, p, as the variable co-ordinates of two 
auxiliary curves, T, and T . p^ will be the. tangents of the angles 
which the elements of these curves make with the axis of p. 

Any assumed value of p^ being substituted in T^ Equation (183), 
will give the tangent of this angle, and this, Equation (184), multi- 
plied by dji^ will give the difference of distances of the ends of the 
corresponding element of the curve from the axis of p. Beginning 
therefore, at the point in which the auxiliary curves cut the axis of 
p, and adding these successive difterences together, a series of ordi- 
nates x and z^ separated by intervals equal to dp^ may be found, and 
the curves traced through their extremities. 

At the point from 
which the projectile 
is thrown, we have, 

x = ; z — ; ^=tan a, 

and the auxiliary 
curves will cut the 
axis of ^, in the same 
point, and at a dis- 
tance from the origin equal to tan a. Let A B, be the axis of p, 
and A C, the axis of x and of z ; take A B z^ tan a, and let BzD, 
and BxE^ be constructed as above. 

Draw the axes Ax and Az, though the point of departure A^ 
-^^g- ^) '•> draw any 
ordinate c z^ x^ to the 
auxiliary curves Fig. 
(I): lay off Ax^ Fig. 
(2) equal to Cx, Fig. 
(1), and draw through 
x^ , the line x^ z, 
parallel to the axis 
Az, and equal to cz^ 
Fig. (1) ; the point 
z, will be a point of 

the trajectory. The range AD, is equal to ED, Fig. (1). 

10 




A 




146 ELEMENTS OF AI^ALYTICAL MECHAlSnCS. 

By reference to the value of (7, Equation (177), it Avill be seen 
that the value of T^ Equation (183), will always be negative, and 
that the auxiliary curve whose ordinates give the values of ic, can, 
therefore, never approach the axis of p. As long as p is positive, 
the auxiliary curve whose ordinates are ^, will recede from the 
axis p ; but when p becomes negative, as it will to the left of 
the axis A (7, Eig. (1), the tangent of the angle which the element 
of the curve makes with the axis p^ will, Equation (185), become 
positive, and this curve will approach the axis p^ and intersect it at 
some point as D. 

The value of p will continue to increase indefinitely to the left 
of the origin A^ Eig. (1), and when it becomes exceedingly great, 
the logarithmic term as v»^ell as (7, and unity may be neglected in 
comparison with p, which will reduce Equations (178) and (179) to 



dp dp 

dx = ; dz = ; 

c.p'^ c.p 



and integrating, 



X = C - —', ^ = (7" + - . log 2?, 
cp c ° ^ 

which will become, on making p very great, 

X = C; z = C" + -log^, 

which shows that the curve whose ordinates are the values of .r, 
will ultimately become parallel to the axis p^ while the other has 
no limit to its retrocession from this axis. Whence we conclude, 
that the descending branch of the trajectory approaches more and 
more to a vertical direction, which it ultimately attains ; and that 
a line G L^ Eig. (2), perpendicular to the axis x^ and at a distance 
from the point of departure equal to C'^ will be an asymptote to 
the trajectory. 

This curve is not, like the parabolic trajectory, symmetrical in 
reference to a vertical through the highest point of the curve ; 
the angles of falling will exceed the corresponding angles of rising, 
the range will be less than double the absciss of the highest point, 
and the angle which gives the greatest range will be less than 45°. 



MECHANICS OF SOLIDS. 



14:7 



Denoting the velocity at any instant by v, we have 



dx^ + d. 



(1 + p^) 



d x^ 



and replacing 
(180), we find 

1 



df \ ' Jr J ^^^ 

and df^ by their values in Equations (178) and 



^.(1 +^2) 



^ c -p yXT^ - log (i? + -v/l +i>') 



(186) 



and supposing ^:) to attain its greatest value, which supposes the 
projectile to be moving on the vertical portion of the trajectory, 
this equation reduces, for the reasons before stated, to 



V c 

which show^s that the final motion is uniform, and that the velocity 

will then be the same as that of a heavy body which has fallen 

1 A'2 



in vacuo through a vertical distance equal to 



2c 



2^ 



§ 152. — When the angle of projection is very small, the projectile 
rises but a short distance above the line of the range, and the equation 
of so much of the trajec- 
tory as lies in the immc- „ 
diate neighborhood of this 
line may easily be found. 
For, the angle of projec- 
tion being very small, jp 
will be small, and its A 
second power may be 
neglected in comparison 
with unity, and we may 
take, 

d 8 — dx\ 

which in Equation (175), gives, 



J) 



and 6- ir a;; 



dp 
dx 



d'^z 

7^ 



2 ex 



2 h . cos^ a 



(187) 



148 ELEMENTS OF ANALYTICAL MECHANICS. 
Integrating, 

2 ex 

dx 4:C .h . cos^ a ' 

making a; = 0, we have — — == tan a, 

whence, 

1 



C = tan a + 



4 c . A . cos^ a 
which substituted above, gives, 

2cx 

dz e I 

= tan a — ; -— -f- 



dx 4:C . h . cos^ a 4:C . h . cos^ a ' 

and integrating again 



€ X 

z = tana,x — -— — — + h C\ 

Sc^ . h . cos^ a 4 c . A . cos^ a 



making x z=z 0, then will z = 0, and 

1 



C" = 



Sc^ .h. cos^ a 
hence, 

z = tanax- '^ --- (e''-2cx - l) . . (188) 

From Equation (IT^)? we have, 

cf .dt^ = — dx . dp^ 
and substituting the value of o?p, from Equation (187), 



, e . dx 

dt — 



-}/ 2gh . cos a ' 
and integrating, making a; == 0, when t — 0, 

(<■"-!) • • • • (189) 



t = . 1 



c \/ 2 f/h . cos a 



MECHANICS OF SOLIDS. 



149 



which will give the time of flight to any point whose horizontal 
distance from the piece is equal to z. 

§ 153. — Let the projectile fall to the ground at the point D, and 
denote the co-ordinates of this point hj x = I, and z = X, and sup- 
pose the time of flight or t = r. These values in Equations (188) 
and (189), give 



8 c2 . A . cos2 a (X — I. tan a) 



cos a .T . c . ■\/2(/h = 



cl 



-2cZ- 1, 
1 . . . 



(190) 
(191) 



When the two constants h and c, as well as a and X, are known, 
these equations will give the horizontal distance Z, and the time of 
flight. Conversely, when the quantities a, I. X and r are known, 
they give the co-efficient of resistance c, and the height A, due to 
the velocity of projection, and therefore, Equation (135), the initial 
velocity itself. 

Eliminating the height A, we find 

- 4 (X - Ltana)(e'^ - \Y^g.rK{e^"^ - 2cl - 1) ; • • (192) 

from which the value of c may be found, and one of the preceding 
equations will give h, or the initial velocity. 

It may be worth while to remark that if the exponential term 
in Equation (188) be developed, and c be made equal to zero, which 
is equivalent to supposing the projectile in vacuo, we obtain Equa- 
tion (ICI). 

LAWS OF CENTRAL FORCES. 



§ 154. — Let a body in motion 
be subjected to the action of a 
deflecting force of attraction di- 
rected to a fixed centre. The 
curve described by the body in 
this case is called an orhii. 

Assume the origin of co-ordi- 
nates at the centre, and denote 



150 



ELEMENTS OF ANALYTICAL MECHANICS. 



the intensity of the attraction on the unit of mass by F^ which we 
will suppose to vary according to any law. Then will 



^ o y 

cos a = ; cos p = ; cos y 

r r ' 



in which r denotes the radius vector of the body ; and Equations 
(119) will, omitting the accents, reduce to 



d^y dP-x 



2/ = 0, 



6?-x 

which being integrated, give 






dt 

dx 

~dT 

dz 
~dJ 



dx 
X ;— . y 



. z 



y 



dt 

dz 
~dt 

dy 
dt 



. X = C'\ 
. z = a'\ 



(193) 



m which C\ C" and C'"^ are the constants of Integration. 

Multiplying each by the first power of the variable which it does 
not contain, and adding, we have, 

C'z J- C"y + C"'x = 0, 

which is the equation of an invariable plane passing through the 
centre, and of which the position depends upon the constants C, 
C", C". Whence we conclude that a moving body deflected to- 
wards a centre, will describe a plane curve. 

§155. — Take the co-ordmate plane xy io coincide with this plane, 
and the Equations (193) will reduce to 



d y d X 

dt dt ^ 



(194) 



MECHANICS OF SOLIDS. 151 

Substituting in Equation (123)', MF for P; dr for c?^ ; making 
P\ P'\ &:c. equal to zero, and recalling that the angles a, /3 and y 
are obtuse, we have, 

3/ F2 4- 2 f MFdr -(7=0.... (195) 

, These two equations will make known all the circumstances of 
the motion. 

§ 156. — But the discussion will be facilitated by transforming 
them to polar co-ordinates ; and for this purpose we have 

X =. r . cos a ; y = r . sin a ; 
differentiating, 

dx =1 dr eos a — r sin ac?a, 
dy =: dr sin a -{- r cos a da. 

Substituting in Equation (194), we find 

-f^'X —.y=r^.---=C':- - • (196) 

dt dt ^ dt ' ^ ' 

integrating again, we have, 

fr-^.da = C't + C", 

and taking between the limits r^ , a^ and r^^ , a^^ , corresponding to 
the time t^ and ^^^ , 

/-'"' rKd^= C'(l„-t,) .... (197) 

But Jr'^da is double the area described by the motion of the 

radius vector; whence we see. Equation (197), that the areas de- 
scribed by the radius vector of a body revolving about a fixed cen- 
tre, are proportional to the intervals of time required to describe 
them. 



152 ELEMENTS OF ANALYTICAL MECHANICS. 

Making, in Equation (197), t^^ — t^ equal to unity, the first mem- 
ber becomes double the area described in a unit of time. Denoting 
this by 2 c, that equation gives 

Placing this in Equation (197), we find 

/ r^ . a a. 



2c 



(198) 



That is to say, any interval of time is equal to the area de- 
scribed in that interval, divided by the area described in the unit 
of time. 

§ 157. — The converse is also true; for, differentiating Equation (196), 
vre find, ' ' 

c?^ ?/ cl X 

Multiplying by M^ and re23lacing M . -7-^ and M. — by their 

values in Equations (120), there will result 

Yx - Xy = 0, 

which is the equation of the line of direction of the force ; and having no 
' independent term, this line passes through the centre. Whence we con- 
clude, that a body whose radius vector describes about any point 
areas proportional to the times, is acted upon by a force of which 
the line of direction passes through that point as a centre. The force 
will be attractive or repulsive according as the orbit turns its con- 
cave or convex side towards the centre. 

§158. — Replacing C by its value 2 c, in Equation (196), and di- 
viding by r^, we have 

s-s <-) 

• The first member being the actual velocity of a point on the 



MECHAN'ICS OF SOLIDS. 153 

radius vector at the distance unity from the centre, is called the 
angular velocity of the body. TJie angular velocitij therefore varies 
inversely as the square of the radius vector. 

§ 159.— Multiply Equation (199) by d s, and it may be put under 
the form, 

ds 2 c 



^ d i r doi '' 

ds 

but — ^- — J is equal to the sine of the angle which the element of 

the orbit makes with the radius vector, and denoting by j) the 
length of the perpendicular from the centre on the tangent to the 
orbit at the place of the body, we have 

r . d (X 
p = r. — — , 
ds 

and 

y=j (200) 

whence, the actual velocity of the body varies inversely as the dis 
tance of the tangent to the orbit at the body's place, from the 
centre. 

§ 160. — Differentiating Equation (195), we faid, 

VdV = - Fdr; 

and taking the logarithms of both members of Equation (200), 

log F = log 2 c — log ^ ; 

differentiating, 

dV dp 

and dividing the equation above by this, 

r^ = ^.,.g = .^l,.|^ (20.) 



154: 



ELEMENTS OF ANALYTICAL MECHANICS. 




GVJL 



AVhence we conclude that, the 
velocity of a body at any point 
of its orbit is the same as that 
which it would have acquired had 
it fallen freely from rest at that 
point over the distance M E^ equal 
to one-fourth of the chord of cur- 
vature M G^ through the fixed cen- 
tre — the force retaining unchanged 
its intensity at M. 

g 161. — To find the differential polar equation of the orbit, we 
have 

~ dJ^ "^ Jt"^ ' 

substituting this in Equation (195), and dividing by M^ 
d 7-2 -{- r"^ d a^ 



dfi 



-\-2fFdr= C'\ 



eliminating dt by Equation (199), we get 



and making r 



-(£+''^)-/-5=^' 



(202) 



differentiating and reducing we have 



^=4o^«^(^-:+«) 



(203) 



From which the equation of the orbit may be found by inte- 
gi-ation when the law of the force is known ; or the law of the 
force deduced, when the equation of the orbit is given. 



MECHANICS OF SOLIDS. 155 

In the first case, the integral will contain three arbitrary con- 
stants — two introduced in the process of integration, and the third, c, 
existing in the differential equation. These are determined by the 
initial or other circumstances of the motion, viz. : the body's velocity, 
its distance from the centre, and direction of the motion at a given 
instant. The general integral only determines the nature of the 
orbit described : the circumstances of the motion at any given time 
determine the species and dimensions of the orbit. 

In the second case, find the second differential co-efficient of u in 
regard to a, from the polar equation of the curve ; substitute this 
in the above equation, eliminating a, if it occur, by means of the 
relation between u and a, and the result will be F^ in terms of u 
alone. 

§ 162. — Let the force vary directly as the distance from the centre; 
required the nature of the orbit. Denote by k the intensity of the 
force on a unit of mass at the unit's distance ; then will 



F=hr :=—' 
u 



and this, in Equation (203), gives, 



+ u 



da?- AiC"" u^ 

Multiplying by 2 c^ w, and integrating, 

7^ -^ u^ =. C - — |-- (204) 

When the radius vector is perpendicular to the orbit, then will 

3- = ; and, therefore, — - = ; 
da, ' ' ' t/a ' 

and denoting the value of the radius vector in this position by y^ , 
and the value of the corresponding velocity by V^ , we have 

4c2 =z F^2^,2. 



156 ELEMENTS OF ANALYTICAL MECHANICS, 

and the value of (7, will be given by 

^ - rs + yz' 

which, substituted above, gives 

d^ - VJY} F/ r,2 zi2 "" ^ 5 

whence, 

1 2«.(?w 



a = 






/ ; 



adding and subtracting under the radical the expression, 

/F/4-^2/<^\2 

the above may be written, 

2r2F2 



1 F 2 _ r 2 k 



2u . d u 



/VLT^IIIjuJ^^ - F/ - r/A:\2 
V ~ V F,^"^=^7^ / 

and integrating, 

^, , , . -i2r2. F2.w2 _ F2 — r2^ 
2{c. + ,) = sm -^ yTTTTTi ^' 

in which 9 is the constant of integration. 

Let the axis from which a is estimated, coincide with the normal 
radius vector ; then, when 



a = 0, will ^l'^ = — - ; 
r 2 



and we have, 



2 9 = sin 1 z= 



which substituted above, gives, 



K« + t) = 



. -1 2r2 F2.^2_ F2 _r2yfc 



/ / 



"" ■ V ,^-r/k 



MECHAl^ICS OF SOLIDS. 157 

and from which we have, 

s,n2 (a + _) = cos 2. = \._,/, ; 

replacing cos 2 a by cos^ a — sin^ a, finding the value of u^. and 
substituting therefor — ? we obtain, after a slight reduction, 



1 



v/ 



1 2 J. ^ • 2 

cos^ a + tT^ • sni^ a 
r/ 



(205) 



2 --- ' ]7 2 



which is the equation of an ellipse referred to the centre as a pole, 
the semi-axes being 

r, and — ^» 

§ 163. — The time required to describe the entire orbit being 
denoted by T^ we have. Equation (198), 

T= ' ^ ^= ^ (206) 

2 

Whence we conclude, that the orbit described by a body under the 
action of a central force which varies directly as the distance from 
the centre, is an ellipse ; and that the time required to perform one 
entire revolution about the centre, is constant, being the same for all 
orbits, great and small, and is dependent solely upon the intensity of 
attraction at the unit's distance. The result of this proposition is of 
the greatest importance in physical science, as we shall have occasion 
to see when we come to the subjects of Acoustics, Optics, &c. 

§ 164. — Let the central force vary inversely as the square of 
the distance: required the orbit. 

Employing the same notation as in the last proposition, we shall 
have 



k u"^ ; 



158 ELEMENTS OF ANALYTICAL MECHANICS, 
which in Equation (203) gives 






multiplying by 2 d u, and integrating 






To determine the constant C, we 
recall that 



du d \r) 
d oi~~ da 



1 



dr 



r^. d oL r. tan s 




in which s denotes the angle made 

by the radius vector with the element of the curve ; and if this be 
known for any radius vector ?'^, corresponding to the place from 
which the body is projected, then will s be the angle of projection 
in reference to the centre, and, 



C=z 



__JL_ + 1 

?• 2 . tan^ s r} 



2Jc 1 21c 



4 c^ r^ ~~ r^ . sin^ s 4 c^ r^ ' 



but, Equation (200), 



whence, 



1 _ Zl - ^/^^/ 

r^2 , sin^ g ~~ 4 c^ ~ 4 c^ r^ 



C = 



F,2 .r,-2k 



in which V^ is the velocity when the radius vector is r^. Substi- 
tuting this above, we get 



du" 
da? 



V^ . r , -2h ^"^ _ ( _ ^ V 

4c^7; ^ 16 c4 ~~ V*"~4^/ 



MECHANICS OF SOLIDS. 159 

whence 

1 — du 

da — 



V 4c2r, "^16 c* / __ y 4 cV 

and integrating 



a + (p = cos 



^^ - 4^ 



^ 16 c* 



V 4^"^ 



The value of 9 is found by the condition that a = 0, when 
7^ — Taking the cosine of both members, replacing il by its 

Value — 5 and reducing, we have 
r 

4 r2 

(207) 



\/-^^^^ ^ + F . cos (a + 9) 



which is the equation of a conic section, the pole being at the 
focus, and the angle (a + 9) estimated from the nearest vertex. 
Comparing it with the equation. 



a (1 - e^) 



14-6. <^os (a -f- 9) 
we find 



(208) 



2 _ {V?r,-^k).^c^ ^ ^ ^ 



but 4 c^ = r^^ . F^2 ^ gjj^2 j^ whence 



A:2 
and 



^ - -^-^ r, . F,2 sin2 £ 4. 1 ; . . . (209) 



ail-e^)^i^^Yllll^. . . . (210) 



160 ELEMENTS OF ANALYTICAL MECHANICS. 

Multiplying both numerator and denominator of the first factor 
in the second member of Equation (209), by M r^, the orbit will be 
an ellipse, parabola or hyperbola, according as 

that is to say, according as the living force of the body, at any 
point of its orbit, is less than, equal to, or greater than twice the 
quantity of work its weight at that point, supposed constant, would 
generate were it to fall freely through the corresponding radius 
vector to the centre. 

And it is a remarkable fact, that the species of conic section is 
w^holly independent of the direction in which the body is projected. 

In the case of the ellipse and hyperbola, the major or transverse 
axis is 

^»=l7¥^2i; ('''> 

which is also independent of the direction of the projection. 

In the case of the parabola, the distance i), from the focus to 
vertex, is given by the equation 

F,2 r 2 . sin^ s 



D = a{l - e) = 



2k 



The position of the transverse axis in reference to the radius 
vector r^ , is obtained by making a = 0, and ?• = r^ ; thus, 

a(l-e2) 1 F2.r.sin2£-^ 

cos 9 = — ^ = —^ '-— 

Vj e € Ice 

Making a + 9 = 90°, the corresponding value of r will give the 
semi-parameter ; that is, 

■• = ^ = k (2'1)' 



MECHANICS OF SOLIDS. 161 

arid because the semi-conjugate axis is a mean proportional between 
the ^emi-parameter and semi-transverse axis, we have, denoting the 
semi-conjugate axis by 6, 

b =z r^ . Vj . sin s . \/-^ (212) 

which depends upon the angle of projection. 

§ 165. — To give an example of the reverse process, let it be 
required to find the law of the force which will cause a body to 
describe a conic section when directed to one of the foci. 

The equation of the orbit referred to the focus, is 

_ « (^ - g^) . 
1 + ^cos a 

whence, 

1 1 + ^ COS a 

r a [I — e^) 

and, 

d^ u — e cos a 
do? a{\ — e^) 

which, substituted in Equation (203), give 

reducing and replacing u by its value — ? we have, 



4c2 ] 

a ( 1 — e-) r'^ 



(213) 



and from which we conclude, that the only law is that of the in- 
verse squaie of the distance. 

§ 166. — If € be made equal to zero, the conic section becomes a 
circle, in which case a = r, and the above becomes 

4c2 

11 



162 ELEMENTS OF ANALYTICAL MECHANICS. 

Also in Equations (199) and (200), we have 
r =z a, and p = a-j 



whence, 



da, 2c , ^^ 2c 
— = — and .V = — ; 
dt a-^ a 



that is to say, both the angular and absolute velocity will be con- 
stant. 

Denoting the time required to perform an entire revolution by 
T — called the periodic time. Then, Equation (198), will 

r=^=af^ ...... (214) 

§167. — Resuming Equations (120), we have 

dx 

^^ ^^ d^x _^ dt 

de- dt 

and performing the operation indicated, regarding the arc of the 
orbit as the independent variable, we have, after dividing both nu- 
merator and denominator by ds"^^ 

dt d'^x dx d^t 



X = M' 


ds 


ds^ ds' ds-^ fds^ d'^x dx ds^ 
dt^ ~^^ Ldt^' ds^ ds ' dt^ 
ds^ 


dH' 
ds\ 


but, 




ds^ d^t d^s ds 

dt^ ds'' ~ df- ' dt ~ ' 





whence, 

^' L ds-' ^ ds dt^J 

In like manner, 

L ds-' ^ ds df A' 

L ds^ ^ ds dfi A 



MECHANICS OF SOLIDS. ' 163 

Squaring and adding, 



"^ ' df^ \ds ' ds'' ~^ ds ' ds^ ^ ds ' ds^y ' 

but, denoting the radius of curvature by p, we have 

(fr')'+C4-»'+©)'-i* 



3P 



and multiplying the second terna of the second member of the 
preceding equation by — ? it may be put under the form, 

M V^ M.d'^s (dx \d'^ X , dy d"- y , dz d"^ z\ , 



1 .d^s rdx d^ X dy d^ y d z d^ z\ , 
~d¥~ V 5 ' ^T^ ^ 11 ' ^d^ '^ ~dl' ^ 1^/ ' 



or, 



^ M F2 M-d^ s * 

2 — — ^— . cos d ; 

p d t^ 

in which o denotes the angle made by the element of the curve and 
radius of curvature ; also 

d x"^ d y^ d z"^ 

whence, substituthig for X- -\- Y"^ -\- Z^ its value BP-^ we have 






and comparing this with Equation (56) we find that R is equal to 
the resultant of the two component forces 

and M - --— , 

p d f- 

* See Appendix No. 2. 



164 ELEMENTS OF ANALYTICAL MECHANICS. 

which make with each other the angle S. But d is equal to OC, 
and therefore 



--^+--(fij)' (-) 



The second of these components is, Equation (13), the inten- 
sity of the reaction of inertia in the direction of the tangent, and the 
first is therefore its reaction in the direction of the radius of 
curvature. 

This first component is called the centrifugal force, and may be 
defined to be the resistance which the inertia of a body in motion 
opposes to ivhatever deflects it from its rectilinear path. It is measured, 
Equation (215), by the living force of the body divided by the radius 
of curvature. The direction of its action is from the centre of 
curvature, and it thus differs from the force which acts towards a 
centre, and which is called centripetal force. The two are called 
central forces. 

§ 168. — If the component in the direction of the orbit be zero, then 
will 

and denoting the centrifugal force by F^, we have 



(216) 



and integrating the next to the last equation, we have 

'n = ""='■' 

in which C is the constant of integration. Whence, the velocity will 
be constant, and we conclude that a body in motion and acted upon 
by a force whose direction is always normal to the path described, 
will preserve its velocity unchanged. 



MECHANICS OF SOLIDS. 



165 



EOTARY MOTION. 

8 169. — Ha\inor discussed the motion of translation of a single 
body, we now come to its motion of rotation. To find the circum- 
stances of a body's rotary motion, it will be convenient to transform 
Equations (118) from rectangular to polar co-ordinates. But before 
doing this, let us premise that the angular velocity of a body is the 
rate of its rotation about a centre. The angular velocity is measured 
by the absolute velocity of a point at the unifs distance from 
the centre^ and taken iji such position as to make that velocity a 
maximum. 

§ 170. — Both members of Equations (38) being divided by d t^ 
slve 



dx' 
dt 


= z' . 


d\ 
dt 


- y' 


d (p 
• dt 


dy' 
dt 


= x' 


d (p 
' dt 


- z' 


d -ztf 
dt 


dz' 




d ttf 




c?4. 


dt 


= y' • 


dt 


— x' . 


dt 



(217) 



in which the first members taken in order, are the velocities of any 
element, as m, in the direction of the axes x, y, z, respectively, tn 
reference to the centre of inertia, § 75, while 



dzi 






cZ(p 
'dV 



are the angular velocities about the same axes respectively. 

Denoting the first of these by v^, the second by v^,, and the third 
by v^, we have 



dzi 

~dJ 



d4. 

~dl 



d (p 



(218) 



166 ELEMENTS OF ANALYTICAL MECHANICS, 

and Equations (217) may be written 



dx' 
~dt 

dt 

dz' 
~dT 



= z .y)y — y.v,, 



x' .v,_ — z' .v^^ )■ 



y .V, 



(219) 



§171. — If an element m be so situated that its velocity shall be 
equal and parallel to that of the centre of inertia, then, for this 
element, will each of the first members qf Equations (219) reduce 
to zero, and 



2 . v„ 



r .^ 



X .V^ Z . Va, 



y' .v^ — x' . Vy 



(220) 



the last being but a consequence of the two others, these equations 
are those of a right line passing through the centre of inertia, 
every point of which will have a simple motion of translation 
parallel and equal to that of the centre of inertia. The whole 
body must, for the instant, rotate about this line, and it is, there- 
fore, called the Axis of Instantaneous Rotation. 

§172.— Denote by a,, 
/8^ , ^^ , the angles which 
this axis makes with the 
CO ordinate axes a:, y, z, 
respectively. Then, tak- 
ing any point on the in- 
stantaneous axis, will, 




cos a, = 



y7'2 ^ yf'z ^ ^r/ 



COS (3 J 



cos 7^ = 



vv2^r7^"+^ 



yVM^/M^^ 



MECHAiN-ICS OF SOLIDS, 
and eliminating x\ y' and z', by Equations (220), 



167 



V\^ + 'y' + 



cos/3^ = 



V\^ + \' + \' 



COS 7^ = 



(221) 



V\^ + v/ + 



which will give the position of the instantaneous axis as soon as 
the angular velocities about the axes are known. 

§ 173. — Squaring each of Equations (219), taking their sum and 
extracting square root, we find 



y 






^' = V(^'-%-y'.^F + G^''-^-^'.vJ2_|_(y'.v^-a:'.v/: 



Replacing v^ , v^ and v^ by their values obtained by simply clearing 
the fractions in Equations (221), this becomes 



V— ■\/m^^ + v^2 _|_ y2 ^ ^j.'2 _j_ y'2 _j_ 2'2 _ (^^' cos a^ + y' cos ^^ -\- z' cos y^y, 

which is the velocity of any element in reference to the centre of 
inertui. 
Making 

x''^ + y"^ + 2'2 = 1, 

we have the element at a unit's distance from the centre of inertia; 
and making 



;' cos a^ + y' cos /3^ + z' cos 7^ — 0, 



(222) 



the point takes the position, giving the maximum velocity. In this 
case V becomes the angular velocity, and we have, denoting the 
latter by v^. , 



'i = V\' + %' + \' 



(223) 



168 ELEMENTS OF ANALYTICAL MECHANICS. 

Equation (222) is that of a plane passing through the centre 
of inertia, and perpendicular to the instantaneous axis. The position 
of the co-ordinate axes being arbitrary, Equation (223) shows that 
the sum of the squares of the angular velocities about the three 
co-ordinate axes is a constant quantity, and equal to the square of 
the angular velocity about the instantaneous axis. 

§ 174.— Multiply Equation (223), by the first of Equations (221), 
and there will result 



(224) 



whence the angular velocity about any axis oblique to the instanta- 
neous axis, is equal to the angular velocity of the body multiplied 
by the cosine of the inclination of the two axes. 

§175. — Equation (223) gives v^. , when v^,v^,Vj, are known. To 
find these, resume Equations (118), and write for the moments of the 
extraneous forces in reference to the axes x^ y,' 2,' through the centre 
of inertia, iV^, M^, L^, respectively, then will 

^"'•W-^ - 71^-0=^^' 'J 

differentiating the first of Equations (219), with respect to t^ we find 

d^x' dz' dy' , dv^^ , cZv 

dt"^ y dt ^ dt dt dt ^ 

^ . d z' ^ d y' ^ . . , . , , , 

and replacmg — — and -7— > by their values given m the second 

and third of Equations (219), 



MECHANICS OF SOLIDS. 



169 



in the same way 






(^2 2 



and these values in the first of Equations, (2"25), give 



Vc/i2 ^^^ ^/ 



- (v^+-£7 •:S77z.2V' ^ = Z, .(226) 

Similar equations will result from the remaining two of Equations 
(225) ; then by elimination and integration, we might proceed to find 
the values of v^ , v^ and v^ , but the process would be long and 

tedious. It will be greatly simplified, however, if the co-ordinate 
axes be so chosen as to make at the instant corresponding to ^, 



^mx' y' — ^\ 1 m z' y' = ', ^ m z' x' — 



(227) 



which is always possible, as will be shown presently. This will 
reduce Equation (22G) to 

^^^ . 2 m (y'2 _f ^'2) _|. ,^ . v^ . 2 m {x'^ - y'^) = X, • 

The other two equations which refer to the motion about the 
axes y' and x\ may be written from this one. They arc, 



(It 

dv 



2 m (.r'2 4- 2'2) 4- v^ . V, . 2 m [z''^ — a:'2) == M^ , 



--.2m (y'2 + ,'2) + v,^ . v^ . 2 m (y'^ - z'^) = N, 



170 



ELEMEJS-TS OF ANALYTICAL MECHANICS. 



The axes x', y\ z\ which satisfy the conditions expressed in 
Equations (227), are called the princijoal axes of figure of the body. 
And if we make 



2 m . (y'2 + a;'2) ^ Q^ 

2 m . (y'2 4- ^'2) =A)} 
we find, by subtracting the third from the second, 

2 m . {z^ - 3/'2) = B — A, 
the first from the third, 

2 m . (^'2 _ x'^) = A - C, 
and the second from the first, 

2m. (/2 _^'2) ^ (7-^5 

which substituted above, give, 



B.^y + v^.v^.{A- C) 



M. 



^.^; + v,.v,.((7-^) = iVr. 



(227)' 



(228) 



By means of these equations, the angular velocities v^ , v^ , v^ , must 
be found by the operations of elimination and integration. 

§ 176. — It is plain that the quantities (7, B and A^ are constant 
for the same body ; the first being the sum of the products arising 
from multiplying each elementary mass into the square of its dis- 
tance from the principal axis z\ the second the same for the prin- 
cipal axis ij\ and the third for the principal axis x' . The sum 
'of the products of the elementary masses into the square of their 
distances from any axis, is called the moment of inertia of the body 



MECHANICS OF SOLIDS. 



in 



in reference to this axis. A, B and C are called principal moments 
of inertia. 

§ 177. — To show that in every body there is a system of rectan- 
gular co-ordinate axes, and in general only one system which will 
satisfy the conditions expressed by Equations (227), assume the for- 
mulas for the transformation from one system of rectangular co- 
ordinates to another also rectangular. These are 

x' = re. cos [x' x) -\- y cos {x' y) -f- z cos {x' z)^\ 

y' — X cos {y' x) + y.cos {y' y) -f 2 . cos {1/ z), \ - - - (229) 

z' = X COS {z' x) + y cos [z' y) + ^.cos {z' z)^] 

in which {x' x)^ W ^) ^^^^ (^'^)' denote the angles which the new 
axes x\ y\ z', make with the primitive axis of x -, (x' y), {y' y) 
and (2'y), the angles which the same axes make with the primitive 
axis of y, and (x' z), {y' z) and {z' z), the angles they make with the 
axis z. 

Assume the common 
origin as the centre of a 
sphere of which the radius 



IS unity ; an 



d conceive the 




points in which the two 
sets of axes pierce its sur- 
face to be joined by the 
arcs of great circles ; also 
let these points be con- 
nected with the point i\", 
hi which the intersection 

of the planes xy and x' y' pierces the spherical surAice nearest to 
that in which the positive axis x pierces the same. Also, let 
^ =z Z' A Z =. X' NX, being the inclination of the plane x' y' to that 

of x y. 
•\> = NAX l)cing the angular distance of the intersection of the 

planes x y and x' y\ from the axis x. 
(p = NAX' being the angular distance of the same intersection 

from the axis x'. 



172 ELEMENTS OF ANALYTICAL MECHANICS 

Then, in the spherical triangle X' JVX, 

cos (x' x) =: cos -^ . cos (p -f Sin -y . sin 9 . cos 6 ; 

In the triangle Y' NX, the side N Y' = ~ + <?, and 

cos (?/' 0^) = — COS -4^ . sin 9 + sin -^ • cos 9 . cos ^ j 

'TT' 

In the triangle Z' iYX, the side iV^Z' = -^5 and 

cos (2' x) = sin n}^ • sin &. 

And in the same way it will be found that 

cos (x' 7/) = — sin 4^ • cos 9 4- cos -|/ . sin 9 . cos 6 ; 
cos {y' y) = sin 4^ . sin 9 + cos -^ . cos 9 . cos & ; 
cos {z' y) = cos 4^ . sin ^ ; 
cos (re' z) = — sin 9 . sin & ; 
cos (?/' z) =z — cos 9 . sin ^ ; 
cos (2' z) = cos ^ ; 

and by substitution in Equations (229), 

x' = X (sin 4/ . sin 9 . cos ^ + cos -^ . cos 9) 

+ y (cos -v]^ • sin 9 . cos ^ — sin -^ • cos 9) — s sin 9 . sin &, 
y' = X (sin "4 . cos 9 . cos & — cos 4^ . sin 9) 

+ y (cos 4^ . cos 9 . cos & + sin 4^ . sin 9) — z cos 9 . sin ^, 
2' = re sin 4' ■ sin ^ + y cos 4^ . sin ^ + ^ cos & ; 

or making, for sake of abbreviation, 

D = X cos 4^ — ?/ sin 4^, 

E = X sin 4^ . cos ^ + y cos 4^ . cos ^ — s sin ^, 

the above reduce to 

re' = ^. sin 9 4- i^ • cos 9, 
y' = ^. cos 9 — i> . sin 9, 
z' = re . sin -4^ . sin d + y . cos 4^ . sin ^ + z . cos &. 



MECHAN"ICS OF SOLIDS. 173 

Substituting these values in the equations 

we obtain from the first, 

sin (p . cos (p . 2 m (^2 _ X)2) + (cos2(p — sin^cp) IjnU.B = 0, 

or, replacing sin 9 . cos 9, and cos^ 9 — sin^ 9, b j their equals ^ sin 2 9, 
and cos 2 9, respectively, 

sin 29. 2 m (^2 _ x)2) + 2cos2 9.2 7?2i).^= 0; • • • (230) 
and from the second and third, respectively, 

cos 9 . 2 m . ^. s' — sin 9 . 2 m i) . 0' = 0, • • • (231) 
sin 9 . 2 m . ^. s' + cos 9 . 2 »z i) . 2' — 0. • • • (232) 

Squaring the last two and adding, we find 

{^m.E. z'Y + {^m.D . z'f = 0. 

which can only be satisfied by making 



Im.U.z' = 0; 
2 m . D . 0' = 0. 



(233) 



These equations are independent of the angle 9, and w^ill give the 
values of -l and 6 ; and these being known. Equation (230) will give 
the angle 9. 

Replacing B and D by their values, we have 

^. 2' = sin ^ . cos & {x^ sln2 -^^ + 2 re y sin 4. cos n^ + y^ cos2 4. — z^) 
+ (cos2 & — sin2 6) (.-c 2 sin 4^ + y ^ - cos 4^) , 

2) . z' = sin Q {xy (cos2 4. — sin2 4.) + (x^ — y^) sin 4. cos 4.} 
+ cos & [xz cos -^ — yz sin 4-) . 

and assuming 

2 m a,-2 := A'; Ijiiy"^ = B'; 2 m z'^ = C ; 
"Lmxy — E' ; J.mxz = F' ; 2myz — //', 

and replacing sin ^ . cos ^, and cos2 <3 — .sin2 ^, by their respective 
values, J- sin 2 ^, and cos 2 ^, Equations (2'33) become 

sin 2 ^ [A' sin2 4^ + 2 E' sin 4. cos + + ^' cos2 4. — C) ^ 



+ 2 cos 2 ^ {F' sin 4. -\- IF cos 4.) 



H 



174 ELEMENTS OF ANALYTICAL MECHANICS. 



sin &{E' . (cos2 4- — sm2 v|.) + {A' — B') . sin v]. cos ^J.} 
+ cos d {F' cos 4. — jy' sin 4) 



Ho. 



in which A\ B\ C\ E\ F' and II\ are constants, depending only 
upon the shape of the body and "the position of the assumed axes 
X, y, z. 

Dividing the first by cos 2 ^, and the second by cos ^, they 
become 



tan2^.(^' sin2 4. + 2 E' sin + cos ^ + 5' cos 2 .). — ^)\_^,^^ 
+ 2 {F' sin -l + H' cos +) | - 5 ( ) 

tan ^ . {^' (cos2 4 — sin2 4) + {A' - B') sin + cos ^^} ) 

+ F' cos 4 - J7' sin 4 ) ^* ^^^^^^ 

From the first of these we may find tan 2 13, and from the second, 
tan ^, in terms of sin 4, and cos 4 J and these values in the equation 

-^^=l4^. ...... (236) 

will give an equation from which 4 niay be found. 
In order to effect this elimination more easily, make 

tan 4 rr ■M, 
whence 

. , u 1 

sm 4/ z=z — — r=r ; cos ■\> 



yT-M^^ yT+1^ 

making these substitutions above, we find 



tan2^=: ^ /v ' 



A'u^ + 2E'iL-\- B' - C'{\ + w2) 



^'^'^ - ~ ^^' (1 _ Z62) + (.r - B') U 

which in Equation (236) give 

L B'F'-F'C'-E'H' ) 1 



MECHAI^ICS OF SOLIDS. 175 

which is an equation of the third degree, and must have at least 
one real root, and, therefore, give one real value for ■^. This value 
being substituted in either of the preceding equations, must give a 
real value for ^, and this with 4^, in either of the Equations (231) 
or (232), a real value for (p ; whence we conclude, that it is always 
possible to assume the axes so as to .satisfy the required conditions, 
and that there is in every body at least one system of 2n-incipal 
axes, at right angles to each other. 

The three roots of this cubic equation are necessarily real ; and 
they represent the tangents of the angles which the axis x makes 
with the lines in which the three co-ordinate planes x' y', y' z\ x' z\ cat 
that Q)i xy \ for there is no reason why we should consider one 
of these angles as given by the equation rather than the others, and 
the equations of condition are satisfied when we interchange the 
axes x' y' z'. Hence, in general, there exists only one set of prin- 
cipal axes. If there were more, the degree of the equation would 
be higher, and would, from what we have just said, give three times 
as many real roots as there are systems.. 

If E' — R' — F' — 0, Equation (237) will become identical ; the 
problem will be indeterminate, have an infinite number of solutions, 
and the body consequently an infinite number of sets of principal 
axes. Such is obviously the case with the sphere, spheroid, &c. 

MOMENT OF INERTIA, CENTEE AND KADIUS OF GYRATION. 

§ 178. — The quantities A, B and C, in Equations (227)' are the 
moments of inertia of the body in reference to the principal axes. 
To fnid these moments in reference to any other axes having the 
same origin as the principal axes, denote by 

x\ y\ z\ the co-ordinates of m referred to the principal axes ; by 
.r, ?/, 2-, the co-ordinates of the same element referred to any 
other rectangular system having the same origin ; and by 
C", the moment of inertia referred to the axis z ; 
then from the definition, 

C — ^ m . {x- + y2) _ 2 ra x"^ -\- I^my^, 



176 ELEMEK'TS OF ANALYTICAL MECHANICS. 

but by the usual formulas for transformation, 

X = ax' + h y' -\- c z\ 
y = a' x' -{- h'y' + c' z\ 
z — a" x' ^ h" y' + c" z\ 

in which a, 5, &c., denote the cosines of the angles which the axes of 
the same name as the co-ordinates into which they are respectively 
multiplied make with the axis corresponding to the variable in the 
first member. 

Substituting the values of x and y in that of C\ and reducing by 
the relations, 

2 m x'y' = 0) 27nx'z' = 0; 2 m / s' = ; 

and we have, 

C" = a"2.2m(?/'2-[-2'2) 4- 6"2.2m(a:'2 + s'2) + c"2 . 2 m (a:'2 + y'2) . 

and by substituting A, B and C for their values, this reduces to 

C = a"^ A + h"^ B + c"2 C . . . . (238) 

That is to say, the moment of inertia with reference to any axis 
passing through the common point of intersection of the principal 
axes, is equal to the sum of the products obtained by multiplying 
the moment of inertia with reference to each of the principal axes, 
by the square of the cosine of the angle which the axis in question 
makes with these axes. 

§ 179. — Let A^ be the greatest, and (7, the least of the moments 
of inertia, with reference to the principal axes ; then, substituting for 
a"2, its value, 1 — 6"2 — c"2, in Equation (238), we have 

C = A -^ h"^ {A - B) - c"^A - C). . . (239) 

By hypothesis, A — B, and A — C, are positive ; therefore, C is 
always less than A, whatever be the value of 6", and c''. 

A<yain, substituting for c"^ its value 1 — a"^ _ j"2 j^i Equation 
(238), we get 

C = C + a"2 (^A - C) + 6"2 (B - C) ... (240) 
and C" must always be greater than C. 



MECHANICS OF SOLIDS. 177 

Whence, we conclude that the principah axes give the greatest and 
least moments of inertia in reference to axes through the same point. 
If A be equal to B, then will Equation (239), become 

C = {1 - c"^) A + c"2 C\ (241) 

and this only depending upon c", we conclude that the moment of 
inertia will be the same for all axes making equal angles with the 
principal axis, z'. The moments of inertia, with reference to all axes 
in the plane x' y', are, therefore, equal to one another. But all the 
axes in the plane x' y\ which are at right angles to one another, 
are, § 175, when taken with z\ principal axes, and we, therefore, 
conclude that the body has an indefinite number of sets of principal 
axes. ■ 

If, at the same time, we have A — B = (7, then will Equation 
(238) reduce to 

C = C = A = B. 

that is, the moments of inertia are all equal to one another, and all 
axes are principal, the Equation (238) being satisfied independently 
of a", 6", c". 

§ ISO. — Resuming Equations, (33), and substituting the values 
of a:, y, 2, in the general expression, 

which is the moment of inertia with reference to any axis, z^ parallel 
to the axis z'. through the centre of inertia, we have 

2 m (:c2 + ,f) =^m[ (x^ + x'f + (y, + yj] 

= 2 m (.r'2 + y'2) ^ ^^2 j^ y2) ,2^ 
+ 2 a:^ . 2 m a;' -|- 2 y^ . 2 in y' j 

but from the principle of the centre of inertia, 

l-mx' = 0, and 2 m?/' — 0; 

whence, denoting by d the distance between the axes z and z\ and 
by M the whole mass, 

\ 
2 w . (a;2 + y2) ^ 2 m {x"^ + y'2) ^ m^?: . . . (242) 

12 



178 ELEMENTS OF ANALYTICAL MECHANICS. 

That is, the moment of inertia of any body in reference to a given 
axis, is equal to the moment of inertia with reference to a parallel 
axis through the centre of inertia, increased by the product of the 
whole mass into the square of the distance of the given axis from 
that centre. 

And we conclude that the least of all the moments of inertia is 
that taken with reference to a principal axis through the centre of 
inertia. 



§ 181. — Denote by r the distance of the elementary mass m from 
7*2 = a;2 + y2^ 



the axis 2, then will 



and 

Now, denoting the whole mass by M^ and assuming 
l.mT^ — MB, 



we have 

/5! r/7. r2 

(243) 






The distance k is called the radius of gyration^ and it obviously 
measures the distance from the axis to that point into which if the 
whole mass were concentrated the moment of inertia would not be 
altered. The point into which this concentration might take place 
and satisfy the condition above, is called the centre of gyration. 
When the axis passes through the centre of inertia, the radius k 
and the point of concentration are called principal radius and prin- 
cipal centre of gyration. 

The least radius of gyration is, Equation (243), that relating to 
the principal axis with reference to which the moment of inertia is 
the least. 

If k^ denote a principal radius of gyration, we may replace 
2 m (x'"^ H- y'"^) in Equation (242) by Mk^^ and we shall have 

2mr2 zz. ifP = M{k;^ + c?2) .... (244) 



MECHANICS OP SOLIDS. 



179 



If the linear dimensions of the body be very small as compared 
with d, we may write the moment of inertia equal to Md"^. 

The letter Jc with the subscript accent, will denote a principal 
radius of gyration. 

§ 182. — The determination of the moments of inertia and radii 
of gyration of geometrical figures, is purely an operation of the cal- 
culus. Such bodies are supposed to be continuous throughout, and 
of uniform density. Hence, we may write d M for w, and the sign 
of integration for 2, and the formula becomes 



2mr2 



/ 



dM.r"^ 



(245) 



Example 1. — A physical line about an axis through its centre and 
perpendicular to its length. 



Denote the whole length by 2 a ; then 
2a:dr : : M:dM, 



whence. 



and 



dM = M- 



2a 



^» 



Mk 



--/: 



M'~ 'dr 

2a 






Ic, = 



-v/3 



If the axis be at a distance d from the centre, and parallel to 
that above, then, Equation (244), 



k = -v/J-a2 + d\ 

Example 2. — A circular plate of uniform density and thickness^ 
about an axis through its centre and perpendicular to its plane. 



180 ELEMENTS OF ANALYTICAL MECHANICS. 

Denote the radius by a; the angle XA Q Y 

by & ; the distance of dM from the centre 
by r; then, 



whence. 



and 



•r a^ : r . d & . d r : : M : d M: 



r.dr. dd 
dM = M' — > 

It a^ 




pa p-i^ A.3 fj « pa «3 

Mh^ = M'-^'d^ = / 2M,^-dr : 

J d (^ a^ do a^ 

k^ = a y^, 

and for an axis parallel to the above at the distance d, 



k = y^ia^ + dK 

Example 3. — The same body about an axis through its centre and 
in its plane. 



As before. 



dM=z if. 



r.dr.dQ 
'jra'^ 



in which r denotes the distance of d M from the centre ; and taking 
the axis to be that from which & is estimated, the distance of the 
elementary mass from the axis will be r sin 6, and 



Mk 



'^ ~Jo Jo 



a pi-It y.3 ^ gj^-^2 ^ 



M 



^dr 



If a^ 



.d& = - — - / r3(] -cos2^)(/r.rf^ 



Mk. 



2 — _ / r^.dr z= M—^ 
' a^ d 4 



and 



and about an axis parallel to the above and at the distance c/, 



= v^ 



4- d-^. 



MECHANICS OF SOLIDS. 



181 



It is obvious that both the axes first considered in Examples 2 
and 8 are principal axes, as are also all others in the plane of 
the plate and through the centre, and if it were required to find 
the moment of inertia of the plate about an axis through the centre 
and inclined to its surface under an angle cp, the answer would be 
given by the Equation (238), 

Mk^^ = -J- Ma^ sin^ 9 + ^ Ma"^ cos^ 9 
= \Ma^{l + sin2 9), 

and for a parallel axis whose distance is d, 

MB = Jf (^i- a2 (1 4- sin2 9) + d^ - 

Example 4. — A solid of revolution about any axis perpendicular to 
the axis of the solid. 

Let D A' E be the given axis, 
cutting that of the solid in A'' L-et 
A' be the origin of co-ordinates, 
P M = y, A' F = x; AA' = m; 
A' B z=z n; and V = volume of the 
solid. 

The volume of the elementary- 
section at F will be 



and 

whence. 



'jf y^ . dx, 

V: M::'!r.7j^.dx: dM-, 

M 

d M = — • It ' y"^ • d X, 




and its moment of inertia about M M\ is. Example 3, 

and about the parallel axis, D E^ 

M 

y.'^.y^.dx{ly^ + x^) 



182 ELEMENTS OF ANALYTICAL MECHANICS, 

therefore, 

nn M 

V m 



But 



whence, 



F 






The equation of the generating curve being given, y may be elimi- 
nated and the integration performed. 

Exam'ple 5. — A sphere about a line tangent to its surface. 
The equation of the generatrix is 

2/2 = 2 a a; — a;2 • 

in which a is the radius of the sphere. Substituting the value of y"^ 
in the last equation, recollecting that m == 0, and ?i = 2 a, we have 



F 



/2a 
{o? x"^ -{- a x"^ — ^ x^) d X 



p2a 5 

/ i^ax — x'^)dx 
Also Equation (244), 

Jc,^ = ^2 _ ^2 _ q2^ 

o 

and 

1c^ =z a 

Centrifugal force arising from the rotation of the earth upon its axis, 
§ (182)'. — If Fi denote the angular velocity of a body about a 
centre, then will V = p Fj, and Equation (216) becomes 



MECHANICS OF SOLIDS. 



183 




The earth revolves about its axis 
A A' once in twenty four hours, 
and the circumferences of the par- 
allels of latitude have velocities 
vhich diminish from the equator to 
the poles. To find the law of this 
diminution, let W be the weight of 
a body on the surface of the earth 
in any parallel of which R\ is the 
radius ; its centrifugal force will be -^ 

W 
9 

in which W is the weight of the body, and Fi, is the angular 

W 

velocity of the earth. Substituting M for — , we have 

F, rr M V,' R'. 

Denoting the equatorial radius C E z= C P^ hj R, an.d the angle 
C P C ~ P C E^ which is the latitude of the place, by 9, we have 
in the triangle P C C", 

R' ~ R cos 9 ; 

which substituted for R' above, gives 

i^, = if Fi^ i2 cos 9 (245)' 

The only variable quantity in this expression, when the same 
mass is taken from one latitude to another, is 9 ; lohence we 
conclude that the centrifugal force varies as the cosine of the latitude. 

The centrifugal force is exerted in the direction of the radius R' 
of the parallel of latitude, and 
therefore in a direction oblique 
to the horizon T T'. Lay off 
on the prolongation of this 
radius, the distance P H^ to 
represent this force, and resolve 
it into two components PN 
and P T, the one normal, the 
other tangent to the surface of 




184 ELEMENTS OF ANALYTICAL MECHANICS. 

the earth ; the first will diminish the weight W by its entire value, 
being directly opposed to the force of gravity, the second will tend 
to urge the body towards the equator. 

The angle HPN is equal to the angle P C E, which is the 
latitude, denoted by 9 ; whence the normal component 



and 



but, 



PN =1 PR X cos 9 =z i^^ . cos 9 = M Vi' R cos2 9, 
PT = PIT sin (p — F^. sin cp = 31 V,^ P . sin 9 cos 9 ; 

sin 9 . cos 9 = 1 sin 2 9 ; 



therefore, 

P 5^ = Ji/Fi^i? sin 29; 

whence we conclude, that the diminution of the weights of bodies 
arising from the centrifugal force at the earth's surface, varies as the 
square of the cosine of the latitude ; and that all bodies are, in con- 
sequence of the centrifugal force, urged towards the equator by a force 
which varies as the sine of twice the latitude. 

At the equator and poles this latter force is zero, and at the 
latitude of 45° it is a maximum, and equal to half of the entire 
centrifugal force at the equator. 

At the equator the diminution of the force of gravity is a 
maximum, and equal to the entire centrifugal force ; at the poles 
it is zero. The earth is not perfectly spherical, and all observations 
agree in demonstrating that it is protuberant at the equator and 
flattened at the poles, the difference between the equatorial and 
polar diameters being about twenty-six English miles. If we sup- 
pose the earth to have been at one time in a state of fluidity, or 
even approaching to it, its present figure is readily accounted for by 
the foregoing considerations. 

The force of gravity which varies, according to the Newtonian 
hypothesis, directly as the mass and inversely as the square of the 
distance from the centre of the earth, is, therefore, on account of a 



MECHANICS OF SOLIDS. 185 

difference of distance and of the centrifugal force, of the earth com- 
bined, less at the equator than at the poles. 

To find the value of the centrifugal force at the equator, make, 
in Equation (245)', M— I and cos 9 = 1, \vhich is equivalent to 
supposing a unit of mass on the equator, and we have 

in which if the known radius of the equator .and angular velocity 
be substituted, we shall find 

F^ := V-.E =z , 1112. 

By the aid of this value, it is very easy to find the angular 
velocity with which the earth should rotate, to make the centrifugal 
force of a body at the equator equal to its weight. 

By the new rate of moticjn, 

cf =^ S2 , 1937 = V/^B; 

f 
in which 32 , 1937 is the force of gravity at the equator. 

Dividing the second by the first, and we find 

32,1937 F/2 ^_ 

-^71112 =lv =289, nearly; 

whence, ' 

that is to say, if the earth were to revolve seventeen times as fast 
as it does, bodies would possess no weight at the equator ; and the 
weights of bodies at the various latitudes from the equator to the 
poles. diminishing in the ratio of the squares of the cosines of lati- 
tude, the weights of all bodies, except at the poles, would be affected. 

IMPULSIVE FORCES. 

§ 183.— We have thus far only been concerned with forces whoso 
action may be likened to, and indeed represented by, the pressure 
arising from the weight of some defuiite body, as a cubic foot of 



186 ELEMENTS OF ANALYTICAL MECHANICS. 

distilled water at a standard temperature. Such forces are called 
incessant, because they extend their action through a definite and 
measurable portion of time. A single and instantaneous eflfort of 
such a force, called its intensity, is assumed to be measured by the 
whole effect which its incessant repetition for a unit of time can 
produce upon a given body. The effect here referred to is called 
the quantity of motion, being the product of the mass into the 
velocity generated. That is. Equations (12) and (13), 

P = if.r, = if<^=ir^; .... (246) 

ill which Vj, denotes the velocity generated in a unit of time. 

The force P, acting for one, two, or more units of time, or for 
any fractional portion of a unit of time, may communicate any other 
velocity F, and a quantity of motion measured by M V. And if 
the body which has thus received its motion gradually, impinge upon 
another which is free to move, experience tells us that it may 
suddenly transfer the whole of its motion to the latter by what 
seems to be a single blow, and although we know that this transfer 
can only take place by a series of successive actions and reactions 
between the molecular springs of the bodies, so to speak, and the 
inertia of their different elements, yet the whole effect is produced in 
a time so short as to elude the senses, and we are, therefore, apt to 
assume, though erroneously, that the effect is instantaneous. Such 
an assumption implies that a definite velocity can be generated in an 
indefinitely short time, and that the measure of the force's intensity 
is, Equation (246), infinite. 

In all such cases, to avoid this difficulty, it is agreed to take the 
actual motion generated by these blows during the entire period 
of their action, as the measure of their intensity. Thus, denoting 
the mass impinged upon by if, and the actual velocity generated 
in it when perfectly free by F, we have 

F = MV ^ M.p^, (247) 

in which P, denotes the intensity of the force's action, and the 
second member of the equation the resistances of the body's inertia. 



MECHANICS OF SOLIDS. 



18T 



Forces which act in the manner just described, by a blow, are 
called impulsive forces. 

MOTION OF A BODY "DISTDEK THE ACTION OF IMPULSIVE FORCES. 

§ 184. — The components of the inertia in the direction of the axes 
xyz^ are respectively 

,, ds dx ,, dx 

M'- — = if- — ; 

d t ds d t 

__ ds dy ,_ dy 

d t ds d t 

__ ds dz ^^ dz 

M'- — — = if- — ; 

dtds dt 

which, substituted for the corresponding components of inertia in 
Equations {A) and (^), give 

dx ^ 

2 F cos a = 2 m • -7- 'j 1 

d t 



2 P cos /3 = 2 m • -^ ; 



(248) 



dz I 



2 P cos 7 

/ dv d x\ 

2 P {x' cos /3 — ?/' cos a) — ^m [x' ' -/- — v' • -f.) ' 

/ , dx , ^A 

2 P {z' cos a — re' cos 7) = 2 m J 2 • — — x- — ) , 

^ . d z , dy\ 

2P(y'cos7 -2'cos/3) z=z 2m {^/ . —— z' - -j^J . 



(249) 



In which it will be recollected that x y z arc the co-ordinates of w, 
referred to the fixed origin, and x' y' z\ those of the same mass 
referred to the centre of inertia. 



MOTION OF THE CENTRE OF INERTIA. 

§185. — Substituting in Equations (248), for dx^ dy, d z, their 
values obtained from Equations (34), and reducing by the relations 

I,mdx' -0) I.md7/-0; 2mdz' -Q-, • • . (250) 



188 ELEMENTS OF ANALYTICAL MECHANICS, 
given by the principle of the centre of inertia, we find 

^ w dx. ^ 

2 P cos a = — ' . 2 w; 
d t ' 

:sFcos(3 z=z J^'Hm; 
(t t 

dz, 

2 P cos 7 = -jf-'Em; 

and substitutino; 31 for 2 m, we have 



d X 
2 P cos a = M-—r^', 
a t 



2Pcos/3 = if.^; 
a t 



(251) 



2 P cos 7 = j)^ 



d z^ 
Tt'' 



which are wholly independent of the relative positions of the elements 
of the body, and from which we conclude that the motion of the 
centre of inertia will be the same as though the mass were concen- 
trated in it, and the forces applied immediately to that point. 

§ 186. — Replacing the first members of the above equations by 
their values given in Equations (41), and denoting by V the velocity 
which the resultant R can impress upon the whole mass, then will 

2P cosa rr JSf Fcos a; 2P cos/3= if Fcos ^ ; 2 P cos/ = Jlf Fcosc; 

substituting these above, we find 

dx, 



V . 


COS a = — -^ 
dt 


V 


cos h = — ^ 
dt 


V 


dz, 
'''''= dl 



(252) 



MECHANICS OF SOLIDS. 



189 



and integrating, 



V-Gosa.t -{- C' 



y^ = V.Qosb.i+C", V (253) 

0, = F.cosc.^l + C"\ ^ 

and eliminating t irom these equations, V will also disappear, and 
we find, 

cos c 



Vr 



y, = 



cos a 

cos c 
cos b 

cos b 



C COS c — 


C" COS a 


cos 


a 


C" cos c — 


C" COS b 


COS 


b 


C COS b - 


C" cos a 
) 



(254) 



which being of the first degree and either one but the consequence 
of the other two, are the . equations of a straight line. This line 
makes with the axes ar, y, 0, the angles a, 5, c, respectively, and is, 
therefore, parallel to the resultant of the impressed forces. 

Whence we conclude, that the centre of inertia of a body acted 
upon simultaneously by any number of impulsive forces, will move 
uniformly in a straight line parallel to their common resultant. 



MOTION AEOUT THE CENTRE OF INERTIA. 

§187. — Substituting, in Equations (249), for dx^ dij and dz^ their 
values from Equations (34), reducing by 

2 7^ a:' = 0, 
2 m y' = 0, 

2 m 2' == 0, 
and we find, 

2 P (a:' cos /3 - y' cos a) =: 2 m (x' . ^ - y' ^-^ ; 

2 P (2' cos oi - x' cos y) = ^m(^z' ' ^ - x' . ^^') ; !- • • (255) 

2 P {y' cos y - z' cos /3) =: 2 m (y' . ^^ - z' . ^') ; 



190 ELEMENTS OF ANALYTICAL MECHANICS. 

whence, the motion of the body about its centre of inertia will be 
the same whether that point be at rest or in motion, its co-ordinates 
having disappeared entirely from the equations. 

AN"GULAE VELOCITT. 

§ 188. — Substituting, in Equations (255), for dx\ dy' and d z' their 
values, which are respectively 

d-\/ .z' z=z ~- da^ .y\ 

d Q^ .X' z=:z — d'ui .z\ 

d'us ,y' =z — d-\^ . x\ 

obtained from Equations (35), (36), (37), and replacing the first mem- 
bers of Equations (255) by ii^,J^^and iV^, respectively, §175, we have 

c?9 



whence 



-^.2m(.:'2-|_y2) ^ L 
0/ t 

d (p L, 

It ~ 2m.(a;'2 -j- y'^y 

d^ _ M, 

dt "" 2 m. {x''^ + s'2) ' 

d-ui _ Nj 

ITt ~ 2 ??i . (2/'2 + z"^) 



(256) 



(257) 



That is, the component angular velocity about either axis, is equal to 
the moment of the impressed forces divided by the moment of inertia 
with reference to that axis. 

ds 

The resultant angular velocity being denoted by -—■> we also 

have 



ji = Tt \A^^ + "^^ + '^'■ 



(258) 



MECHANICS OF SOLIDS, 



191 



§ 189. — The axis of instantaneous rotation is found as in § 171, 
bj making in Equations (217), 

dx' - 0; dy' = 0; d z' = 0; 
which gives, 

z' = f.4^;z' = x^-^;y' = x'.4^; . . (259) 
which are the equations of a right line through the centre of inertia. 



AXIS OF SPOXTAJ^OrS KOTATION. 

§ 190. — If both members of Equations (34) be divided by d t^ we 

have 

d X d Xj d x\ 

d t d t d t 

dt d t d t 

d z dz^ d z\ 

dt d t dt 



and if for any series of elements we have 

dy 



d X 

dt ' dt 



r. dZ 



(260) 



then will 



dx, 
'~dt 



d x' , dy, d y' , d z, 



dz' 



d t dt 



dt dt 
dz' 



d t 



(261) 



, . . . dx' dy' ^ ^ ^ - . 

and substitutino; for -— -, -f— and — , — , their values given m 
° dt dt dt ^ 

d X d 1/ d z 

Equations (217), and for -— ^? -— ' and —- ^) their values given by 

Ct C (JL L • Of C 

Equations (252), we find 

, f d cp V . cos a. 

d t 
V . cos 6 



dsi 



d zs 
~dT 
, d-\^ V . cos c 



y — X . -— — 
^ d-a 



d •KT 

IT 



(262) 



192 ELEMENTS OF ANALYTICAL MECHANICS. 

Which are the equations of a right line parallel, Equations (259), to 
the instantaneous axis. 

This line is called the axis of spontaneous rotation ; because, being 
at rest, Equations (260), while the centre of inertia is in motion, the 
whole body may be regarded, during impact, as rotating about this 
line. Its position results from the conditions of Equations (261), 
which are, that the velocity of each of its points, and that of the 
centre of inertia must be equal and in contrary directions. The dis- 
tinction between the axes of instantaneous and of spontaneous rota- 
tion is, that the former is in motion with the centre of inertia, while 
the latter is at rest. 

The relative positions of the axis of spontaneous rotation, and 
the line along which the resultant impact acts, cannot be affected 
by any change in the co-ordinate axes. For simplicity, take the 
fixed axes so that the movable axis x' shall be parallel to the 
line of the resultant, and the plane x' y^ shall contain this latter line. 
In this case, 

cos a rr 1 ; cos 6 = 0; cos c = ; 

and the values of y' and z\ in L^ M^ N^ will be, 

y' = e,- z' =.0- 

in which e^ is the perpendicular distance from the centre of inertia 
to the direction of the resultant impulse. 
Also, in Equations (254), we shall have, 

cos c ^ cos b 



cos a 



in Equations (257), 



d(p —Re, 



dt 2m. (.t'2 4- y'2) » 

d-L ^ d'ui 



and, Equations (262), 



d(p d(p 

d^ ' dzi 



MECHANICS OF SOLIDS. 193 

whence, the axis of spontaneous rotation is perpendicular to the 
direction of the resultant impulse. 

From the first and second of Equations (262), we have hy clear- 
ing the fractions, 



- V 



dt ^ dt 

, dzi , c?9 

and since (f 4. = ; d-m = ; we obtain, after substituting the value 

of ^, that of B = M F, and that of 2 tti . {x'^ + y'^) = Mk,^, 
d t 

a;' = 0; y'=-^' (263) 

The axis of spontaneous rotation is, therefore, in the 'plane z' y\ 
and cuts the perpendicular drawn from the centre of inertia to the 
line of the resultant, and at a distance from that centre equal tc 
the square of the principal radius of gyration with reference to the 
axis of instantaneous rotation, divided by the distance of the line 
of the resultant from the same centre. 

Adding e^ to both members of the second of the above equations 
without respect to signs, we have, after writing e for y\ 

e-\-e,= "' ^'' =1 (264) 

in which I is the distance from the axis of spontaneous rotation to 
the line of the resultant impact. 

g 191, — The body being perfectly free, and the axis of spontaneous 
rotation at rest while the other parts of the body are acquiring 
motion, it is plain that the forces, both impressed and of inertia, are 
so balanced about that line as to impress no action upon it. The 
points of the body on the line of the resultant impulse are called 
centres of percussion in reference to the axis of spontaneous rotation. 
A centre of percussion is, therefore, any point at which a body may 
be struck in a direction perpendicular to the plane through the centre 

13 



194: ELEMENTS OF ANALYTICAL MECHANICS. 

of inertia and axis of spontaneous rotation, without communicating 
any shock to a physical line coincident in position with that axis. 

§ 192. — In Equation (258), we have 

dt dt ~ ^m (a;'2 4- y''i) h^ ^ ' 

§ 193, — If T denote the distance of the elementary mass m, from 
the axis of z\ the velocity of rotation of this element will be 

and its centrifugal force 

dap" r^ m d ©2 

dt'' r dt'' ' 

which is the pressure exerted by the inertia of m upon the axis z', 

this axis being that of instantaneous rotation. Its point of application 

is on that axis. The cosines of the angles which its direction makes 

x' y' 

with the axes x' and y' are respectively — and — ; its components 

in the direction of these axes, are, therefore, 

-^'x'-m and ^-/-m, 

and its moments in reference to the axes y' and x\ are 

jj-m^xz and j^-m^y ., 

and the sum of the moments of the centrifugal forces of all the 
elements of the body in reference to these axes are 

-^ .2m- x' z' and -^ -^.m-y' z' , 
di^ dt' ^ 

Now if the instantaneous axis be also a principal axis, then wiU 

^m.x' z' = 0, and 2 m y' 2' = ; 
and there will be no pressure on the instantaneous axes. If, there- 



MECHAXICS OF SOLIDS. 



195 



fore, the impressed force be so applied as to cause the body to 
begin to rotate about a principal axis, the rotation will continue 
about this axis, and the axis is said to be permanent; but if the 
rotation do not begin about a principal axis, the axis of rotation 
will change its position under the pressure arising from the centrifu- 
gal forces developed, and this change will continue till the position 
of the axis of rotation reaches that of a principal axis. 

MOTION OF A SYSTEM OF BODIES. 



§194. — We have seen that the Equations (117) and (119) give 
all the circumstances of motion of the centre of inertia of a single 
body in reference to any assumed point taken as an origin of co- 
ordinates. For a second, third, and indeed any number of bodies, 
referred to the same origin, we would have similar equations, the 
only difference being in the values of the co-ordinates, of the inten- 
sities and directions of the forces, and of the magnitudes of the masses. 
This difference being indicated in the usual way by accents, we should 
obtain by addition, 






2 M 



d?z 
~d¥ 



=z :e. z 



(266) 



:2m(x 


d'^y 
dt'^ 


V',;)=nr''-Xy); 


2 M {z 


d^x 
de 


''■'/^)=H^^-z^); 


^M{y 


d-'-z 
dt'' 


— S) = M^.-^^); 



(267) 



in which it must be recollected that x, y, z, &c., denote the co- 
ordinates of the centres of inertia of the several masses iV, ^c, 
referred to a fixed origin. 



196 



ELEMENTS OF ANALYTICAL MECHANICS. 



MOTION OF THE CENTBE OF INEKTIA OF THE SYSTEM. 

§195. — Taking a movable origin at the centre of inertia of the 
entire system, denoting the co ordinates of this point referred to 
the fixed origin by x^ , y^ , 0^, and the co-ordinates of the centres 
©f inertia of the several masses referred to the movable origin by 
x\ y\ z\ &c., we have, the axes of the same name in the two sys- 
tems being parallel, 

X = X, -\- x\ 

y ^y,^ y\ 



and. 



(268) 



(Px = d^ x^ 4- d"^ x\ 
d? y z=z d? y ^ -\- d"^ y', 
d^z -d"^ z,^ d'^ z\ 

which substituted in Equations (266), and reducing by the relations, 
2Jf.£Z2a;' = 0; 2if(^2y' = 0; 2i/'(^2^'r=0; • -(269) 
obtained from the property of the centre of inertia, we find 



til 
di^ 

d^ 

df^ 

d^z^ 

IF 



j:M- 2: z 



(270) 



which being wholly independent of the relative positions of the several 
bodies, show that the motion of the centre of inertia of the system 
will be the same as though its entire mass were concentrated in 
that point, and the forces applied directly to it. 



§196. — Multiplying Ihe first of Equations, (270), by y^, the second 



MECHANICS OF SOLIDS. 



197 



hj x^, and taking the difference ; also, their first by z^ the third 
by ar^, and taking the difference, and again the second by ^^, the 
third by y^, and taking the difference, we find 



(.. 









:') .2jf =:r,.2F-y,.2X; ^ 






(271) 



df^ ' df^ 

which will make known the circumstances of motion of the common 
centre of inertia about the fixed origin. 

MOTION OF THE SYSTEM ABOUT ITS COMMON CENTKE OF INERTIA. 

§ 197, — Substituting the values of x^ y, 2, d^ x, &c., given by 
Equations (268), in Equations (267) and reducing by Equations (271) 
and (269), there will result 



2 M- (x' 



2jf 



d^y' 


df' 


d?-x' 


df 


d^z' 



d-^x' 



-) 

d'^z' \ 

772/ 



^{Yx' -Xy') 
2 {Xz' - Zx') 



■('■'S^-S)-<./-K.,, 



(272) 



Equations from which all traces of the position of the centre of 
inertia have disappeared, and from which we conclude that the 
motion of the elements of the system about that point will be the 
same, whether it be at rest or in motion. These equations are 
identical in form with Equations (118); whence we conclude that 
the molecular forces disappear from the latter, and cannot, there- 
fore, have any influence upon the motion due to the action of the 
extraneous forces. 

CONSERVATION OF THE MOTION OF THE CENTRE OF INERTIA. 

§ 198. — If the system be subjected only to the forces arising from 
the mutual attractions or repulsions of its several parts, then will 



2X 



2F=0; 2Z = 0. 



198 ELEMENTS OF ANALYTICAL MECHANICS. 

For, the action of the mass M, upon a single element of JiT, 
•will vary with the number of acting elements contained in Jf; 
and the effort necessary to prevent M' from moving under this 
action will be equal to the whole action of M upon a single element 
of M' repeated as many times as there are elements in M' acted 
upon ; whence, the action of M upon M' will vary as the product 
M3f. In the same way it will appear that the force required to 
prevent M from moving under the action of M\ will be propor- 
tional to the same product, and as these reciprocal actions are 
exerted at the same distance, they must be equal ; and, acting in 
contrary directions, the cosines of the angles their directions make 
with the co-ordinate axes, will be equal, with contrary signs. Whence, 
for every set of components F cos a, F cos [3, F cos /, in the 
values of 2 X, 2 Z", 2 Z, there will be the numerically equal com- 
ponents, — F' cos a', — P'cos/3', — F' cos 7', and. Equations (270), 
reduce, after dividing by 2 M, to 

S = «^ T;t = 0; g = o. . . . (..3) 

and from which we obtain, after two integrations, 

x^ = C .t + D'; "^ 

y^ =r (7'W + i>" ; \ ' • • • • • (274) 

z^ = C" t + D'";. 

in which C", C'\ C"\ D\ D" and D'" are the constants of inte- 
gration ; and from which, by eliminating t^ we find two equations of 
the first degree between the variables x^^ y,^ z^^ whence the path 
of the centre of inertia, if it have any at all, is a right line. 

Also multiplying Equations (273) by 2dx^^ 2c??/^, 2dz^^ respec- 
tively, adding and integrating, we have 

<WAJll+Jll ^ y. ^ C .... (275) 

in which C is the constant of integration and V the velocity of the 
centre of inertia of the system. From all of which we conclude, 
that when a system of bodies is subjected only to forces arising 



MECHANICS OF SOLIDS. 



199 



from the action of its elements upon each other, its centre of inertia 
will either be at rest or move uniformly in a right line. This is 
called the conservation of the motion of the centre of inertia. 



COXSEEYATION OF AEEAS. 

§ 199. — The second member of the first of Equations (272) may 
be written, 

Yx' - Xy' + y x" - Xy" + &c. ; 
and considering the bodies by pairs, we have 

X = - X'; T = -Y'-, 
and eliminating X and Y' above by these values, we have 

Y{x' -x") -X{y' -y") + &c. 
But, 



p ■ p ' 

in which p denotes the distance between the centres of inertia of 
the two bodies. And substituting these above, we get 
v' — y" x' — x" 

and the same being true of every other pair, the second members 
of Equations (272), will be zero, and we have 

d^y' , d^x' 



and integrating 






,^ x' d y' — y' d x' ^ , 

^^ -IT =^' 

dt 



(276) 



200 ELEMENTS OF ANALYTICAL MECHANICS. 

But § 156, x' dy' — y' d x[ is twice the differential of the area swept 
over by the projection of the radius vector of the body M^ on the 
co-ordinate plane x' y', and the same of the similar expressions in 
the other equations, in reference to the other co-ordinate planes; 
whence, denoting by A^^ A^, A^, the whole areas described in any 
interval of time, t, by the projections of the radius vector of the body 
M, on the co-ordinate planes, x^ y\ x' z\ and y' z' ; and adopting 
similar notations for the other bodies, we have 

d A^ 

dt ' 

dt ' 

in which C, C", C" ^ denote the sums of the products obtained by 
multiplying each mass into twice the area swept over in a unit of time 
by the projection of its radius vector on the planes x' y' ^ x' z\ y' z' ; and 
by integrating between the limits t^ and t\ giving an interval equal to t, 

^M.A^ = C'.t', 
^M.Ay= C" t', 
^M.A^^ C"'t; 

whence we find that when a system is in motion and is only sub- 
jected to the attractions or repulsions of its several elements upon 
each other, the sum of the products arising from multiplying the 
mass of each element by the projection, on any plane, of the area 
swept over by the radius vector of this element, measured from 
the centre of inertia of the entire system, varies as the time of the 
motion. This is called the principle of the conservation of areas. 

§ 200. — It is important to remark that the same conclusions 
would be true if the bodies had been subjected to forces directed 
towards a fixed point. For, this point being assumed as the origin 
of co-ordinates, the equation of the direction of any one force, say 
that acting upon JH/, will be 

Yx- Xy =.0- 



MECHANICS OF SOLIDS'. 201 

and the second members of Equations (267) will reduce to zero ; 
and the form of these equations being the same as Equations (272), 
they will give, by integration, the same consequences. 

mVAKIABLE PLANE. 

§201. — If we examine Equations (276), we shall find that M- -j— 

is the quantity of motion of the mass M^ in the direction of the 
axis y\ and is the measure of the component of the moving force 

dx' 
in that direction ; the same may be said of M' ——'> in the direc- 
tion of the axis x' ; whence the expression, 

,^ x' d y' — y'd x' 
at 

is the moment of the moving force of M^ with respect to the 
axis z'. Designating, as before, the sum of the moments with respect 
to the axes z\ y' and x\ by L,^ M^^ N^^ respectively, Equations (276) 
become 

L,= C ; M,^ C" ■ N, = C". 

Denoting by A^ B and (7, the angles which the resultant axis 
makes with the axes z\ y' and x\ we have, § 110, 



cos A = 



cos B 



cos C = 



Vi/' 


M, 


+ N,^ 


^L} 




+ N,^ 



C^ 

C" 



(277) 



These determine the position of the resultant or principal axis. 
The plane at right angles to this axis is called the principal plane. 
The position of this plane is invariable, and it is therefore called 
the invariable plane^ either when the only forces of the system are 
those arising from the mutual actions and reactions of the bodies 
upon each other, or when the forces are all directed towards a fixed 
centre. 



202 ELEMEN-TS OF ANALYTICAL MECHANICS. 



PBINCIPLE OF LIVING FOKCE. 

§ 202. — The components of the extraneous force, in the direction 
of the axes, impressed upon the element m of a mass if, are 

Pcosa = X; Fcos^=Y; Poos'/ = Z; 

the components of the inertia of m in the directions of the same 
axes are 

d'^x d?y dp- z 

and the resultant components in these directions are 

,^ d? X __ d'^y ^ d'^z 

^-""•Jfi' ^-""-dW' ^-^-7^5 

and their virtual moments, 

and similar expressions for the other elements m\ m'\ &;c. But 
these must be in equilibrio ; whence. 



2 



{ (x-..%) ..+ (r-./^) .,+ (Z-..^) ..( =0; 



in which S x, S 7/ and Sz, are small spaces described by m, in the 
direction of the axes, consistently with the connection of the parts 
of the system one with another at the time t. 

But the spaces actually described at the end of the time t, being 
consistent with the connection of the parts of the system one with 
another, we have, 

Sx = ^.5t; Sy=%.U- Sz = -^^St', 
dt ' ^ dt ^ dt ^ 

which in the above equations give, after transposition, 

{ dx d'^x dy d'^y dz d^ z ) ^ { ^ dx ^ ^^dy ^ ^ dz\ 



MECHAN"ICS OF SOLIDS. 203 

which becomes, by integrating, 

\ dt^ ^ dt^ dfi ) «/ V dt ' dt dt / ' 

or replacing the first member by its value, 

--=-/(-^>^g+4>' (-«) 

§ 203. — If P be the mutual pressure of two elements m and m', 
in contact^ at the point whose co-ordinates are a?, y, 0, then the ex- 
pression 

/(^•£+^^-^-f+-4:)^- • • • (^^^) 

for the element m becomes 

/_^ / dx r> dy c?A 7 

F ( cos a • -; h cos p • -^ + cos y - -—) dt ; 
V dt ^ dt ' dty ' 

and for m', it becomes 

/ri / dx ^ ^ dy ^ dz\ , 

and their sum will be zero ; therefore the pressure P will disappear 
from Equation (278). 

If the elements m and m' be not in contact, but be separated 
by the distance r, let x 7j z and x' y' z' be their respective co-ordinates, 
P their mutual action, supposed some function of r ; then will 

X — x' y — y' z — z' 

cos a = ; cos p == ^ ; cos 7 = ; 



/ ^ — ^ o, y — y , z — z' 

cos a' = ; cos p = — ■ ; cos 7 = 



and the expression (270) for the element m, will be 

fj^fx—x' dx y~y' dy z — z' dz\ 



204 ELEMENTS OF ANALYTICAL MECHANICS, 
and for the element m\ 

/T^{x — x' dx' y — y' dif z — z' dz'\ 
^ y-v- ■ -JT + —r- ■ HT + —r • TT/ '''' 

and F will appear in Equation (278) under the form, 

and since 

7-2 = (a: - xj + (y - 2/')' + (2 - ^OS 
by differentiating 

rdr = {x - x')d{x - x') -\- {y - y')d{y - y') + {z - z')d{z - z')-, 
and the above reduces to ^fPdr^ which in Equation (278), gives 

^mv'^ zzz'l^JP dr (280) 

Any force which acts upon a fixed point, will not appear in the 
equation of the living force, since the velocity of such point is zero. 
Neither will the force of rolling friction, where one of the bodies 
is fixed. The force of sliding friction will. 

If the forces act upon none of the elements of the system except 
such as remain invariably connected during the motion, the living 
force must remain the same throughout the motion, for in this case 
dr ■=. 0, will give. Equation (280), 

2 m «;2 = C. 

This is called the 'principle of the conservation of living force. 

The value of P, being by hypothesis a function of r, will 
always be integrable. The integration being performed, and r being 
replaced by its value in functions of x y z^ x'y' z', and the same for 
r' &c., the living force of the system will become a function of the 
co-ordinates of the bodies' places, and when taken between limits 
will be dependent alone upon the co-ordinates of the first and last 
positions of the bodies, and wholly independent of the paths described 
by them. 



MECHANICS OF SOLIDS. 205 

§ 204. — The co-ordinates of the element w, at the end of the 
time ty being x^/z, we have 

2;2 = ^ {dx^ + dy^ + dz^), 

and substituting for dx, c?y, dz, their values obtained from Equa- 
tions (268), 

v^ = -^ {dx,^ + dy^^ + dz^^) J^^^{dx^dx' + dy, dy' ^ dz, dz') 

substituting this in Equation (278) and reducing by the relations, 

2mdx' = 0] 2mdy' = 0; 2mdz' — 0, 
obtained from the property of the centre of inertia, we find 

2m2;2 — F,22m + 2m2;'2. (281) 

in which V^ denotes the velocity of the centre of inertia of the entire 
system, and v' that of each element about that centre. Whence, the 
living force of a material system in motion is equal to the living 
force arising from the motion of translation of the centre of inertia, 
increased by the living force arising from the motion about the 
common centre of inertia of the whole. 

§ 205.— Differentiating Equation (280), we have 

<i^P,r = i^^.dt; ....'. (282) 

and if, at any instant during the motion the living force become 
a maximum or minimum, then will 2 Fdr = 0, and the system will, 
Equation (28), be in equilibrio. 

Also, § 134, when the living force is a maximum, the position 
which the system assumes would be that of stable equilibrium, if all 
the velocity were destroyed ; and when the living force is a mhiimvm^ 
the position would be one of unstable equilibrium. And since a fimc- 
tion passes through its maximum and minimum values alternately, as 
the variable increases continuously, the system when in motion 



ELEMENTS OF ANALYTICAL MECHANICS. 

will pass through the positions of stable and unstable equilibrium 
alternately. 

§ 206. — If, during the motion, two or more bodies of the system 
impinge against each other so as to produce a sudden change in their 
velocities, the sum of the living forces will undergo a change. To esti- 
mate this change, let A, B^ C be the velocities of the mass w, in the 
direction of the axes before the impact, and a, 6, c what these veloci- 
ties become at the instant of nearest approach of the centres of 
inertia of the impinging masses, then will 

A — a, B — b, C — c, 

be the components of the velocities lost or gained by m at the instant 
corresponding to this state of the impact, and 

771 (A — a), 7n{B — b), m ( (7 — c), 

the components of the forces lost or gained. The same expressions, 
with accents, will represent the components of the forces lost or 
gained by the other impinging bodies of the system. These, by 
the principle of D'Alembert, § 71, are in equilibrio, whence 

^m(A - a)dx -^ :Em{B — b) §7/ -\- :Em{C — c)dz = 0. 

The indefinitely small displacements S x, St/, S z, &c., must be made 
consistently with the connection by virtue of which the velocities are 
lost or gained ; but as a, b, c denote the components of the actual 
velocities of any two bodies during the time of greatest compression, 
when alone these velocities are equal, this condition will be fulfilled 
if we make 

5x =z a.S t; St/ =z b .§ t; S z = c .S i. 

These values being substituted in the above equation, we have, 
after dividing by d t, 

27n(A — a)a-i-Jl7n{B — b)b-\-2m(C—c)c = • . (283) 
or, 

2m (Ja 4- -56 + Cc) - 2m (a2 + &2 _|. c2) = • • (284) 



MECHANICS OF SOLIDS 207 

But we have the identical equation, 

or, 



Aa + jBb + Cc = 



A^ -^ B^ -\- C^ o? + 52 



2 ' 



which in Equation (284) gives, 
2w(^2_^^24.(72)_2^(a2^52_|_c2)^2m[(^-a)2 + (^-6)24-(C-c)2], 

and making 

^2 ^ ^2 ^ (72 ^ F2, 
a2 + 62 + c2 = '^2^- 
2 ^72 _ 2 wi ?/2 ^ 2 m [ (^ - a)2 + (5 - 6)2 + ((7 - c)2] . . (285) 

whence we conclude, that the difference of the sums of the living 
forces before the collision^ and at the instant of greatest compression, 
is equal to the sum of the living forces which the system would have, 
if the masses moved with the velocities lost and gained at this stage 
of the collision. 

Since all the terms of the preceding equation are essentially 
positive, it follows that at the instant of nearest approach of the 
impinging bodies, there is a loss of living force. 

If the impinging masses now react upon each other in a way to 
cause them to be thrown asunder, and A\ B\ C", &c., denote the 
components of the actual velocities, in the direction of the axes, at 
the instant of separation, then will the components of the velocities 
lost and gained while the separation is taking place, be 

a — A\ b — B\ c — C", &c., &c. ; 

and Equation (283) will become 

^m{a — A') a + 2m{b — B') b -\- Im (c — C')c :zz 0, 
or, 

2m(a2 + 62 + c^) - Im (A' a + B' b -f C c) = Oj 



208 ELEMENTS OF ANALYTICAL MECHANICS. 

and eliminating A' a -}- £' b -{- C c, by means of the identical equa- 
tion, 

^2 _|_ 52 _j_ c2 4- ^'2 _|_ ^^2 

we obtain, 



(a - Ay +{b- By + {c- C'f = I 



and making 



{a-Ay^ 
\-{b^B'f K 

L + (^- cy] 



2mz*2_ 2^^7/2^ _2m[(a-^')2 + (i-^')^ + (c~ Cy] -.(286) 

All the terms of this equation being essentially positive, it fol- 
lows, from the sign of the second member, that during the reaction 
of the bodies by which they are separated, there is a gain of living 
force. 

If the loss and gain of velocities after, be the same as before 
the instant of greatest compression, then will there be no loss or 
gain of living force by the collision. 

The principles of Equations (285) and (286) find an important 
application, in the construction, adjustment, and motion of machinery. 

SYSTEM OF THE WOELD. 

§ 207. — The most remarkable system of bodies of which we have 
any knowledge, and to which the preceding principles have a direct 
application, is that called the solar system. It consists of the Sun^ 
the Planets^ of which the earth we inhabit is one, the Satellites of the 
planets, and the Comets. These bodies are of great dimensions, are 
spheroidal in figure, are separated by distances compared to which 
their diameters are almost insignificant, and the mass of the sun is 
so much greater than that of the sum of all the others as to bring 
the common centre of inertia of the whole within the boundary 
of its own volume. 

These bodies revolve about their respective centres of inertia, 
are ever shifting their relative positions, and our knowledge of them 



MECHANICS OF SOLIDS. 209 

is the result of computations based upon data derived from actual 
observation. 

Kepler found ; 

I. That the areas swept over by the radius vector of each planet 
about the sun, in the same orbit, arc proportional to the times of dc' 
scribing them. 

II. That the planets move in ellipses, each having one of its foci in 
the sun^s centre. 

III. That the squares of the periodic times of the planets about the 
sun are proportional to the cubes of their mean distances from that 
body. 

These are called the laws of Kepler, and lead directly to a 
knowledge of the nature of the forces which uphold the planetary- 
system. 

§ 208. — The first law shows, § 157, that the centripetal forces 
which keep the planets in their orbits, are all directed to the sun's 
centre ; and that the sun is, therefore, the centre of the system. 

The second law shows, § 165, that the law of the centripetal 
force is that of the inverse square of the distance. 

Denoting by T, the periodic time of any one planet; by a and 6, 
the semi-axes of its orbit, we have. Equation (198), 



area of ellipse if ab 



and substituting the values of b and c, Equations (212) and (211)' 

2'rr.a 



2'7r.a 2 



whence, 



and for another planet 



rp2 

03 


4':r2 
- k ' 


^2 

a/ 


14 



210 ELEMENTS OF ANALYTICAL MECHANICS, 

but by Kepler's third law, 





r£2 rp2 

a3 ~ a3^ ' 


and therefore 






k,=k, 



whence we conclude that not only is the law of the force the 
same for all the planets, but the absolute force is the same; and that 
the same cause acts upon all the planets ; and that if the planets 
were at the same distance from the sun, the unit of mass of each 
would experience the same intensity of attraction. 

From these consequences of the laws of Kepler, it is inferred that 
the particles of the sun attract those of the planets, and vice versa, 
with a force varying directly as the mass of the attracting particle, 
and inversely as the square of the distance. And from the experi- 
ments of Dr. Maskelyne, who found by observations on the fixed 
stars, that the mountain Shehallien in Scotland drew the plumb line 
sensibly from its vertical position; and also from the experiments 
of Cavendish and Baily upon leaden and other balls, it is inferred 
that this power of attraction resides in every particle of matter, 
wherever found, and that it is exerted under all circumstances without 
the possibility of being intercepted. It is therefore concluded that 
matter is endowed with a general gravitating principle by which every 
particle attracts every other particle, and according to the law above 
mentioned. 

But, according to this principle, not only does the sun attract the 
planets, but the planets attract the sun and one another. Either 
Kepler's laws cannot, therefore, be rigorously true, or universal gravi- 
tation is not a Principle of Nature. Now in point of fact, observa- 
tions of far greater nicety than those of Kepler, prove that his laws 
are not accurately true, though they differ but slightly from reality; 
a circumstance arising entirely from the fact of the great mass of 
the sun as compared with the sum of the masses of all the planets. 
Were there but a single body in existence besides the sun, it would 
describe accurately an elliptical, parabolic or hyperbolic orbit about 



MECHANICS OF SOLIDS. 211 

the common centre of inertia, depending upon its living force and 
the smi's attraction. A third bodj would derange this motion and 
cause a departure from the regular path, and the degree of the 
disturbance would depend upon the mass, distance and direction of 
the disturbing body as compared with those of the sun. The same 
remark would apply to a fourth, fifth, and to any number of addi- 
tional bodies. These disturbances, by which any one body of the 
system is made to depart from the simple path due to the sun's 
action alone, and which are caused by the combined action of all 
the others, are called perturbations. These have been computed, and 
the complete harmony which is found to subsist between the numeri- 
cal results deduced from theory and observation, is the strongest 
possible evidence in support of the Law of Universal Gravitation. 

If the principal plane of the solar system, as determined at 
different and remote periods, be found to have undergone no change, 
this will show that the system is uninfluenced by the action of the 
fixed stars and other distant bodies, and its centre of inertia will, 
§ 198, either be at rest or be moving uniformly through space in 
a right line ; but if the principal plane be found to change its 
place^ it will be a sign that the system is in motion, and that its 
centre of inertia is describing a curvilinear path about some distant 
centre. 

IMPACT OF BODIES. 

§ 209. — When a body in motion comes into collision with another, 
either at rest or in motion, an impact is said to arise. 

The action and reaction which take place between two bodies, 
when pressed together, are exerted along the same right line, per- 
pendicular to the surfaces of both, at their common point of contact. 
This arises from the symmetrical disposition of the molecular springs 
about this line. 

When the motions of the centres of inertia of the two bodies 
are parallel to this normal before collision, the impact is said to be 
dire'^t. 

When this normal passes through the centres of inertia of both 



J- 






212 ELEMENTS OF ANALYTICAL MECHANICS. 

bodies, and the motions of these centres are along that line, the 

impact is said to be direct and 

central. 

When the motion of the centre 
of inertia of one of the bodies is 
along the common normal, and the 
normal does not pass through the 
centre of inertia of the other, the 
impact is said to be direct and 
eccentric. 

When the path described by the 
centre of inertia of one of the bodies, 
makes an angle with this normal, 
the impact is said to be oblique. 

When two bodies come into col- 
lision, each will experience a pres- 
sure from the reaction of the other ; and as all bodies are more or 
less compressible, this pressure will produce a change in the figure 
of both ; the change of figure will increase till the instant the bodies 
cease to approach each other, when it will have attained its maximum. 
The molecular spring of each will now act to restore the former 
figures, the bodies will repel each other, and finally separate. 

Three periods must, therefore, be distinguished, viz. : 1st., that 
occupied by the process of compression ; 2d., that during which the 
greatest compression exists ; 3d., that occupied by the process, as 
far as it extends, of restoring the figures. The force of restitution 
must also be distinguished from the force of distortion ; the latter 
denoting the reciprocal action exerted between the bodies in the 
first, and the former in the third period. 

The greater or less capacity of the molecular springs of a body 
to restore to it the figure of which it has been deprived by the 
application of some extraneous force when the latter ceases to act, 
is called its elasticity. 

The ratio of the force of distortion to that of restitution, is the 
measure of a body's elasticity. This ratio is sometimes called the 
co-efficient of elasticity. When these two forces are equal, the ratio 



MECHANICS OF SOLIDS. 



213 




is unity, and the body is said to be perfectly elastic; when the 
ratio is zero, the body is said to be 7ion- elastic. There are no bodies 
that satisfy these extreme conditions, all being more or less elastic, 
but none perfectly so. 

Let the two bodies AB and A' B'^ the former moving along the 
line H T^ and the latter along 
H' T\ come into collision at the 
point 0. Through 0, draw 
the common normal N L. De- 
note the angle H G N by 9, 
and H' EN by 9' — these being 
the angles which the directions 
of the two motions make with 
the normal. Also denote the 
velocity and mass of the body 
^^ by F and M respectively, and the velocity and mass of A' B' 
by V and M'. 

The components of the quantity of motion of the two bodies in 
the direction of the normal and of the perpendicular to the normal, 
will be 

M V cos (p, if' V cos cp' and M V sin 9, M^ V sin 9', 

The former of these components will alone be involved in the 
impact ; for if the bodies were only animated by the latter, they 
would not collide, but would simply move the one by the other. 
For simplicity, let the body ^ ^ be spherical ; the normal will 
pass through its centre of inertia. 

Denote by w, the velocity of the body A B in the direction of 
the normal at the instant of greatest compression, and by u' the 
velocity of the body A' B' at the same instant in the same direction. 
Then will 



V cos 9 — w, and V cos 9' — u' 



(287) 



be the velocities lost and gained in the direction of the normal, and 
M{Vco^ 9 - w), and J/' ( 7' cos 9' - u') • • • (288) 



/ 



214: ELEMENTS OF ANALYTICAL MECHANICS. 

be the forces lost and gained at the instant of greatest compression ; 
and hence, 

if ( F cos 9 — 7z) -J- M' ( F' cos cp' — u') =z 0; • - (289) 

and denoting the angular velocity of the body A' B^ by F/, the 
distance G' D from the centre of inertia of A' B' to the normal 
by e, and the principal radius of gyration of A' B\ with reference 
to the instantaneous axis by 'k^ , then will 

_ if(Fcos(p - u).e _ (Fcos9-^)g 

y, - -^—^ - -^^ . . (^yu) 

and since the velocity u must be equal to that of the point D at 
the end of the lever arm e, we have 

u = u' -\- e.V; (291) 

Substituting the values of u and u' from this equation successively 
in Equation (280), we find 

if F cos ^ + ilf' V cos 9' + if' e F/ ,^^^, 

^' = - m-Vm' • (^^^> 

, _ ^ ^ cos 9 + if' y COS ^' - Me V/ 

After the instant of greatest compression, the molecular springs 
of the bodies will be exerted to restore the original figures, and 
if c denote the co-efficient of elasticity, then will the velocities lost 
by ^^ and gained by A' B' during the process of restitution be, 
respectively, 

c {V cos cp — u) and c {V^ cos 9' — u') ; 

and the entire loss of AB, and gain of A' B\ will be, respectively, 

Fcos 9 — w + c ( F cos 9 — u), and V cos 9' — u' -\- c{V' cos 9'— u'). 

Also the gain of angular velocity of the body A' B\ during the 
process of restitution, will be 

^^ , ( Fcos 9 — w) . e 

cv: = c — ^^ — , 



MECHANICS OF SOLIDS. 215 

and the whole angular velocity produced by the impact and denoted 
by Vj, will be given by the equation, 

T-r /-. , \ ( ^cos (p —u) e 

^, = {^+<:)- ^— ^ (294) 

Denoting the velocities of A B and A' B', after the collision by 
V and v', and the angles which the directions of these velocities 
make with the normal by & and &', respectively, then ^ill 

i; cos <5 = Fcos(p — F cos 9 -f u—c {V cos (p— u) = {l -f c) u — c Fcos9, 

v'cos &'— P^'cos9— F'cos(p'+w'— c(F'cos(p'— w')=:(l+c)z^'— cF'cos9', 

and replacing the values of u and ■w', as given by Equations (292) 
and (293), 

, ,, , 3/Fcos9 + lf' F'cos^'+if'e F/ 
vcosd = {l-j-c) M-^M^ ^-cFcos<p, (295) 

r ., ,,, ,l/Fcos9+i/' F' COS (p'-ife F/ ^^, , ,^^^, 
v/cos^' = (l + c) 1 -—I ^ — c F'cos9'(296) 

Moreover, because the effects of the impact arising from the compo- 
nents of the quantities of motion in the direction of the normal will 
be wholly in that direction, the components of the quantities of 
motion before and after the impact at right angles to the normal will 
be the same, and hence 

V sin & = Fsin 9, (297) 

v' sin &' = F sin 9' (298) 

Squaring -Equations (295) and (297) and adding; also Equations 
(296) and (298) and adding, we find after taking square root, and 
reducing by the relations 

cos2 & + sin2 & z= I', cos2 6' 4- sin2 ^' =: 1 ; 
^=Y [(1+0 j^j- _^ j^; ^-cFcos9]2H- F^srn^. (299) 



^.A(l+,)^^Z^^+f^;7^^-^^^/-cFW9l^ F^sin^9'-(300) 



216 ELEMENTS OF ANALYTICAL MECHANICS. 

Dividing Equation (297) by Equation (295). and Equation (298) by 
Equation (296), we have, 

V . sin 9 
(1 + c) ^-^-^, ^ c rcos, 

F'.sinffl' 

tan 4' = rp-TT , ,^, T7/ / yir ir r p02) 

^if Fcos® + J/' F cosffl' — if e F/ ^^. , ^ •' 
(1 + .) ^-^-^, c F cos / 

Equations (290) and (292), will give the values of u and F/, in 
known terms, and these in Equations (294), (295) and (296) will 
give the values of F^, v^ and v\ and all the circumstances of the 
collision will be known. 

§ 210. — If the bodies be both spherical, then will e = 0, and Equa- 
tion (294) gives F, = 0; and Equations (299) and (300), (301) and 
(302), become 

.^y/[(l+c)J^l ^+f;;' '^"-^' -cFcos.p+F^sin^, ■ • • (303) 



.'=y[(l+c) -^^^'-7+/J^°-^ -cF'cos,T+ F-sin».' • • (304) 

F sin (p , ^ X 

tan 5 — ji^ , -i^, ^yr, -. • • (30d) 

, J/Fcos (p 4- M' V cos ffl' ^^ ^ ' 

(1+c) i v , ,„ ^-cFcos(p 

^ ^ ^ i/ + if' ^ 

^^-^ ^' = ,, , ,i frcosI+'y' F^^ — - — ; • • (^*'*') 

The Equations (303) and (304) will make known the velocities, 
and (305) and (306) the directions in which the bodies will move, 
after the impact. 

Now, suppose the body A' B' at rest, and its mass so great that 

the mass of AB is insignificant in comparison, then will F' be 

M 
zero, M' may be written for M -{- M' and -^tj- will be a fraction so 



MECHANICS OF SOLIDS. 



217 



small that all the terms into which it enters as a flictor may be 
neglected, and Equation (303) becomes 



V z= V -y/ C^ C0S2 ^ _|_ s^;^2 (p . 



and Equation (305), 



tan 



tan 9 



(307) 



The tangent of ^ being negative, shows that the angle NHK^ 
which the direction of ^4^'s motion 
makes with the normal N N' after the 
impact, is greater than 90 degrees ; in 
other words, that the body AB is 
driven back or reflected from A' B'. 
This explains why it is that a cannon- 
ball, stone, or other body thrown ob- 
liquely against the surface of the earth, 
will rebound several times before it 
comes to rest. 

If the bodies be non-elastic, or, which is the same thing, if c be 
zero, the tangent of 6 becomes infinite ; that is to say, the body 
A B will move along the tangent plane, or if the body A' B' were 
reduced at the place of impact to a smooth plane, the body AB 
would move along this plane. 

If the body were perfectly elastic, or if c were equal to unity, 
which expresses this condition, then would Equation (307) become 




tan 



tan 9 



(308) 



which means that the angle N HF = EHN' becomes equal to 
KHN'. The angle EHN' is called the angle of incidence, tho 
angle KHN\ commonly, the angle of reflection. Whence we see, 
that when a perfectly elastic body is thrown against a smooth, hard, 
and fixed plane, the angle of incidence will be equal to the angle 
of reflection. 



If the angles 9 and 9' be zero, then will cos 9 =: 1, cos 9' = 1, 



218 ELEMENTS OF ANALYTICAL MECHANICS. 

sin 9 = 0, sin 9' =r ; the impact will be direct and central, and 
Equations (303) and (304) become 

, .^ , ^MV-\-M'V' ^., 

and passing to the limits, non-elasticity on the one hand and perfect 
elasticity on the other, we have in the first case, c = 0, and 

MV -\- MY' ,„^^, 

^ = -FT-^^ ('''> 

, MY ■\-MY' ,„,^, 

^ =-^T-^^ ("^'^ 

and in the second, c = 1, consequently, 

if F + J[f' F' 

^ = ^-ifT^^ — ^ (^") 

.. = 2^^fZ:_r' (312) 

M -^ M' ^ ' 



CONSTEAnTED MOTION. • 

§211. — Thus far we have only discussed the subject oi free motion. 
We now come to constrained motion. 

Motion is said to be constrained when by the interposition of 
some rigid surface or curve, or by connection with some one or 
more fixed points, a body is compelled to pursue a path different 
from that indicated by the forces which impart motion. 

§ 212. — The centre of inertia of a body may be made to con- 
tinue on a given surface, by causing it to slide or roll upon some 
other rigid surface. 

§ 213. — We have seen, § 128, that the motion of translation of 
the centre of inertia, and of rotation about that point, are wholly 



MECHANICS OF SOLIDS, 



219 



independent of one another, and the generality of any discussion 
relating to the former will not, therefore, be affected by making, 
in Equation (40), 

§(p = 0; S-^ = 0; (^^ = 0; 
vhich will reduce that equation to 

(2 P COS a — -— - ''Em)S Xj 

Making 

2mr=if; 2Pcosa=:X; 2Pcos/3=F; SPcos/r^Z; 
and omitting the subscript accents, we may write 

Now, assuming the movable origin at the centre of inertia, and 
supposing this latter point constrained to move on the surface of 
which the equation is 

L = F{xyz) =^ 0, (314) 

the virtual velocity must lie in this surface, and the generality of 
Equation (313), is restricted to the conditions imposed by this cir- 
cumstance. 

Supposing the variables x y z, in the above equations, to receive 
the increments or decrements S x, (Jy, S z, respectively, we have, from 
the principles of the calculus, 



dZ 



§x + 



dL . _^dL . 
dy dz 



0. 



dx - ' d. - - ^. - - - P15) 

Multiplying by an indeterminate quantity X, and adding the product 
to Equation (313), there will result 



\ dt^ ^ dx/ 



+ (---S + -S)^^ 



220 ELEMENTS OF ANALYTICAL MECHANICS. 

The quantity X, being entirely arbitrary, let its value be such as to 
reduce the coefficient of one of the variables S x, (5" y, 6 z, say that of 
Sx, to zero; and there will result 

and 

Now in Equation (315), (J y and 5z may be assumed arbitrarily, and 
6x will result; hence S y and 5z in Equation (317) may be regarded 
as independent of each other, and by the principle of indeterminate 
coefficients, 



and eliminating X by means of Equation (316), we find, 

(r-if.^).-^-(x-if.— )~ = o, 



(318) 



(319) 



which, with the equation of the surface, will determine the place of 
the centre of inertia at the end of a given time. 



MOTION ON A CURVE OF DOUBLE CURVATUEE. 

§214. — If the centre of inertia be constrained to move upon 
two surfaces at the same time, or, which is the same thing, upon 
a curve of double curvature resulting from their intersection, take 

L = F(xyz) = 0, ) 



MECHANICS OF SOLIDS, 



221 



from which, by the process of differentiating and replacing dx, dy^ dz^ 
by the projections of the virtual velocity, 



dL ^ dL dL 

dx dy "^ dz ' 

dH . ^dH . ^dH . 

dx dy dz 



(321) 



. (322) 



Multiplying the first of these by X, and the second by X', adding the 
. products to Equation (313), and collecting the coefficients of ^ar, 6y^ 
and 6 z, we have 



/^ _^ d' x ^ ^ dL , _ d H\ . 
(X-J/.— + X._ + X'.— )^. 

dy ' dyJ 



+ (^-^-?l + ^-- + ^' 



\ df" dz d z y 



:0 . (323) 



Now the coefficients of two of the three variables d x, d y and d z^ 
say those of ^a; and S y, may be made equal to zero by assigning 
proper values for that purpose to the indeterminate quantities X and 
X', in which case, since 00 is not equal to zero, its coefficient must 
also be equal to zero ; whence 



X-M.^ + X.'-^ 
a t^ d x 



X'. 



dH 

dx 



0, 



dt" dy dy 



Z - M -^ 4- X — X' ^ 
df- ' d z d z 



= 0. 



.(324) 



and eliminating X and X', there will result 

\ df') ' \dz*"dy dy ' dz y 

^V dt^/ \dx dz dz dxJ ^ ^ ' 






'd L dj£ 

~dx 



d z dx 

dL dJI\ 
dx dy / 



222 



ELEMENTS OF ANALYTICAL MECHANICS. 



which, with the equations of the surfaces, is sufficient to determine 
the co-ordinates of the centre of inertia when the time is given. 

§ 215. — If the given surfaces be the projecting cylinders of a 
curve of double curvature, then will Equations (320) become 



H^F'{yz) 



^0.) 



(326) 



And because L is now independent of y, and H is independent of x^ 
we have 



which reduce Equations (324) to 
X 



^^ d? X d L 



dt^ dz dz 



(327) 



and Equation (325) to 



V d f- / d X dy_ 



dL dH 

z dy 

dL dH 

z 

dL dH 



y= 0. 



(328) 



This, with the equations of the curve, will give the place of the 
centre of inertia at the end of a given time. 

§216. — If the curve be plane, the co-ordinate plane a; 2, may be 
assumed to coincide with that of* the curve; in which case the 
second of Equations (327), becomes independent of y, that varia- 
ble reducing to zero, and 

(^2y = 0, and 1^ = 0: 
dy 



MECHANICS OF SOLIDS. 



223 



hence Equations (327). bcome 



d"^ X d L 

dv- d X 



Y= 
Z 

and because the factor 



^, d^z , ^ dL ^, dH ^ 

dv- dz dz 



il%^ = 0, 
df- ' 



;329) 



d TT 
Equation (328) becomes, on dividing out the common factor — — . 



V dt~y dz \ dt-J ^ 



dL 



0. . (330) 



§ 217. — By transposing the terms involying X, in Equations (316) 
and (318) and squaring we have 






dr~ z\ 2 



l + (---^0 



The second member of this equation is, Equation (.50), the square of 
the intensity of the resultant of the extraneous forces and the forces 
of inertia. Denoting this resultant by iV, we may write 

and dividing each of the equations 
dL 



-"7=-(---"S). 

= -(>-~-"S). 



dL 
dy 
dL 
~d~z 



224 



ELEMENTS OF ANALYTICAL MECHANICS. 



obtained by the transposition just referred to, by Equation (331), 
we find, 



dL 

d X 



X - Jf 



d^^ 

dt^ 



dL 



N 



dt^ 

N 



di^ 



N 



(332) 



The second members are the cosines of the angles which the 
resultant of all the forces including those of inertia, makes with the 
axes ; the first members are the cosines of the angles which the 
normal to the surface at the body's place makes with the same axes. 
These being equal, with contrary signs, it follows not only that the 
forces whose intensities are 

/ {dLX^ , /dzy , /d LY ^ ,^ 



are equal, but that they are both normal to the surface, and act m 
opposite directions. The second is the direct action upon the surface; 
the first is the reaction of the surface. 

Equation (331), will, therefore, give the value of a passive 
resistance sufficient to neutralize all action in the system which is 
inconsistent with the arbitrary condition imposed upon the body's 
path. If the body be constrained to move on a rigid surface or 
line, this resistance will arise from its reaction. 

§ 218. — If Equations (332) be multiplied by 
and the angles which the normal resistance of the surface makes with 



MECHANICS OF SOLIDS. 225 

the axes rr, y, z, respectively, be denoted by d', d" and &'", those 
equations will take the form 






X - M • —~ -\- N' cos y = 0', 



dv- 



(333) 



Z - M' ^^ -{■ J}^'Cos&'" =0. 



§ 210. — To impose the condition, therefore, that a body in motion 
shall remain on a rigid surface, is equivalent to introducing into 
the system an additional force, which shall be equal and directly 
opposed to the pressure upon the surface. Tlie motion may then 
be regarded as perfectly free, and treated accordingly. The same 
might be shown from Equations (324) to be equally true of a 
rigid curve, but the principle is too obvious to require further 
elucidation. 

Equations (333), may, therefore, be regarded as equally appli- 
cable to a rigid curve of any curvature, as to a surface ; the nor- 
mal reaction of the curve being denoted by iV, and the angles 
which iV makes with the axes x, y, z, by 6', 6" and &"\ 

§ 220. — To find the value of iV", eliminate d t from Equations 
(333), by the relation 

2_ _ _r 

dt ~ ds' 

in which V and s are the velocity and the space ; then by transpo- 
sition these equations may be written 

ds^ 

JV^.cos^" = M' F2.^ - F; 
ds^ 

N'Qos^"' = M' F2~ - Z, 
ds^ 

15 



226 ELEMENTS OF ANALYTICAL MECHANICS. 

Squaring, adding and reducing by the relations 

C0S24' + C0S2^'' + C0S2 4''' = 1, 



and we find 



iV^ = 






- 2if. F2 



Resolving i2 into two components, one parallel and the other per- 
pendicular to the path, the former will be in equilibrio with the 
inertia it develops in the direction of the curve ; and denoting 
by 9 the inclination of R to the radius of curvature, we have 



or, 



R,..,^-M.% = M.y^.%, 



H.sin.-M.V^.^^; 



Squaring and subtracting from the equation above, there will result, 



iVr2=: 
but 



^'""^ -R'dJ ^ R'Ts ~^ R'd~s'^ 



multiplying the second member by p -'- p, substituting above, and 
reducing by the relations, 



d-'x 


d— £— 

dxd'^s ds d'^y dy d?s ds 




d'^z dz dh 


4 

ds 


ds^ 


dsds'~ ds' ds^ dsds'~ ds' 




ds^ dsds"^' 


" ds 




dx dy 

X ^Ts Y ^ds 
^^'^ = B'^ ds^R' ds^ 


z 

R 


4* 

ds 






*See Appendix No. 2. 









MECHANICS OF. SOLIDS. 



22T 



and 



ds^ 



P = 



'^(d'^xf + {d'^yf + {d'zf - (d^sy 



in which p denotes the radius of curvature, we have, 



JV2 __ j/2 _ 2 i2cos(p 4- i22cos2(p; 

P P 



and taking square root, 



I^ = 



MV^ 



R cos 9. 



(334) 




The first term of the second member is, 
§ 167, the centrifugal force arising from 
the deflecting action of the curve, and the 
last term is the normal component of the 
resultant R. As the equation stands, its 
signs apply to the case in which the body- 
is on the concave side of the curve, and 

the resultant acts from the curve. The angle 9, must be measured 
from the radius of curvature, or that radius produced, accordmg as 
the body is on the concave or convex side of the curve. When 
the body is moving on the convex side of the curve, the first 
term of the second member must change its sign and become 
negative. 



§221.— Writing Equations (333) under the form 

= X+ iV^cos^', 
d'^y 






M 



M- 



df' 
d'^z 
dt'^ 



F+ iV^cos^", 
Z + iVcos^"'; 



multiplying the first by ^dx^ the second by 2c?y, the third by 2c?«, 
adding and reducing by the relation 



ds 



{dx ., , dy 

( — • cosr + -J- -cos 



ds 



4- -7- • cos 
ds 



'-) . = 0, 



228 ELEMENTS OF ANALYTICAL MECHANICS. 

the second factor being the cosine of the angle made by the nor^^ 
mal and tangent to the curve, we have 

M' (^^— -^ df ) =K^^^+^^1/+Zdz); 

integrating and reducing bj 

' F2 - ^^'+ ^y' + dz^ 
~ dt^ 

we find 

ilf F2 r= 2 f{Xdx ^ Tdy + Zdz) + a • • (335) 

This being independent of the reaction of the curve, it can have no 
effect upon the velocity. 

If the incessant forces be zero, then will 

X=0; ]r = 0; and 2=0; 
and 

that is, a body moving upon a rigid surface or curve, and not acted 
upon by incessant forces, will preserve its velocity constant, and the 
motion mil be uniform. 

We also recognize, in Equation (335), the general theorem of 
the living force and quantity of work ; and from which, as before, 
it appears that the velocity is wholly independent of the path de- 
scribed. 

Example 1. — Let the body be required to move upon the interior 
surface of a spherical bowl, under the action of its own weight. In 
this case, 

X = a:2 -h y2 4. ^2 _ ^2 ^ ; .... (336) 

dL ^ dL ^ dL ^ 

dx dy ^ \ dz * 



HECHAKICS OF SOLIDS 



229 



and the axis of z being vertical and 
positive downwards, 

which values in Equations (319), 
give 






dt^ 



dt^ 



-(337) 




and differentiating the equation of the 
sphere twice, we have 

xd'^x 4- yd'^y + ^.d^z = — {dx^ + dy^ + dz^); 

dividing by df^, and replacing the second member by its value F^, 
the velocity, we find, 

d^x , d^y , d'^z m 



di^ 



df^ 



dt'' 

But, Equation (335), 

V^ = 2gz + C (338) 

and denoting by V and A:, the initial values of V and g, respectively, 
we have 

which substituted above, gives 



d?x d^y d'^z 



dO- 



de- 



d(^ 



<2g {k -z) ^ F'2 . . (339) 



Eliminate a:, y, d'^x^ d'^y^ from this equation by means of Equa- 
tions (336) and (337). 

From the latter we find, 



d^y 
dO' 



y /d^z \ 



d^x 
d 



X X /d^z \ 

F ^ T \dF ■" ^/ 



230 ELEMENTS OF ANALYTICAL MECHANICS. 

which substituted in Equation (339), and reducing hj means of 
Equation (336), we get 

multiplying by 2dz, and integrating, we find i 

a2. ^ = 2ff{a^z - ^3 + kz^) - V' z'^ + C; 

in which C is the constant of integration, and to determine which, 
we denote the component of the velocity F', in the direction of the 
axis Zy by V^\ and make z = k. This being done, we get 

(7 = a2.F,'2 + F2P - 2(fa^ky 
whence, 

a^'^=:2g{aH-z^ + kz^) - V'^z' + a^ F/2 + F'^F - 2ga'k, 

adding and subtracting a^ V^ in the second member, this reduces to 

«2. IJ- = (a^ - 2^) [F'^ -2<^ (i - ^)] - e„ 

in which 

C, = (a^ - k'') F'2 - a2 F/2. 

Finding the value of d I, and integrating, we have 

, • • • • (340) 

-/(a^ - ^2) [F'2 -2ff{k - z)] - C, 

Could this equation be integrated in finite terms, then would z 
become known for a given value of t ; and this value of z in 
Equation (336), and the first of Equations (337), after integration, 
would make known the values of x and y, and hence the position 
of the body ; its velocity would be known from Equation (335). 
But this integration is not possible. 



MECHANICS OF SOLIDS. 231 

g222.' — We may, however, approximate to the result when the 
initial impulse is small and in a horizontal direction, and the point 
of departure is near the bottom of the bowl. Let & be the angle 
which the radius drawn to the variable position of the body makes 
with the axis of ^ ; cp, the angle which Ijhe plane of the angle 6 
makes with the plane through the axis z and initial place of the 
body, supposed in the plane xz -, V = (3 -^/ga, the velocity of pro- 
jection in a horizontal direction, (3 being a very small quantity ; 
and a the initial value of ^. Then, because a is very small, 

k = a cos a, = a (cos^ ^ a — sin^ ^a) = a — ^aoi^; 
and for the same reason, 

2 = a — -J a . fl2 . also, y = x tan <p ; 
F/2 = 0; C, = [a2 _ a^ (1 - f a2)"]./32^a = a^gu^fS^ 
after neglecting ia* in comparison with unity, 

dt dtdz A ^ ^ 

d& ~ dz' d6 ~ ' dz^ 

and substituting the value of the last factor from Equation (340), 

dt fa & /«.,v 

T-A=-\ , • • (341) 

"which may be put under the form 

fa n —4d.d& 



2i = 



9 J ^(^2 _ ^2)2 _ [2^2 _ (a2 + /32)]5 

whence by integration 



2^ =\/-.cos-i 



S^±^i±^y,., . . (34.) 



making t = 0, and 5 = a, we have (7 = — cos 1 . -y/a ■— -^9, or 
(7=0; and solving the equation with reference to ^, we get 

&^ = i (a2 + /32) + J (a2 _ /32).cos2 Y^l . ^. . . . (343) 



232 ELEMENTS OF ANALYTICAL MECHANICS. 

From which it appears that the greatest and least values of d, 
will occur periodically, and at equal intervals of time. The former 
of these values is found by making 

cos 2 \ /- • if = 1; whence 2\/-'t = 0, or = 2 if, or = 4 "T, 
V a \ a ' 

and so on; and for a single interval between two consecutive maxi- 
ma, without respect to sign, 

' = * vf ' (^^*) 

the maximum being a. 

The least value occurs when 

cos2\/^.^ = — 1, or 2\/--^ = cr, or ^Sflf, &c. 
V a V a 

whence for a single interval between any maximum and the succeed- 
ing minimum, 

' = i*V7; (345) 

the minimum being /3. 

The movement by which these recurring values are brought about, 
is called oscillatory motion; that between any two equal values is 
called an oscillation ; and when the oscillations are performed in 
equal times, they are said to be Isochronous, 

Again, 



d& dt db 



substituting for — — , its value obtained from the relation y = xtan(pj 



we find 



dcp 



dcp \ C ^y dx\ dt 

u "^ x^ + 2/2 ' V * "ZT ~ ^ ' TT/ * T^' 



d& x^ -\- y^ 
Integrating the first of Equations (337), we get 



d V d X ^ ^^, r> 7 

at at 



MECHANICS OF SOLIDS. 233 

substituting this abo7e, and also the value of -— - , given bj Equa- 
tion (341), we find 

dS ~ 4 y^(a2 — ^2) ^^2 _ ^2j 
dividing this by Equation (341), 

d cp fq a. /3 [q a . 

Ji 



(346) 



t V a ' d2 ~V a ' 



\ {a? 4-/32)+^ [a? - (3^) . cos2 a/- • t 



but 



whence 



cos 



2^.t = cos^.^.t-sin^y/l.i; 



d(p fg c6 • ^ 



dt V a 



a2.cos2-v/4--i; + /32.sin2 
from which we find 



^i-' 



■ (347) 



/3 



^ -dt 



cos2 \ / — • ^ 



<^9 — - ^ , 



1 4-4.tan2W-:l-.^ 

integrating, and taking tangents of both members, 

tancp = — . tan \/— .; (348) 

from which the azimuth of the plane of oscillation may be found 
at the end of any time. 

Making tan (p = od, we have 

7 1 3 5 

-.^r=-^; or = -^; or = -^cr, &c., 



234: ELEMENTS OF ANALYTICAL MECHANICS, 

and the interval from the epoch to the first azimuth of 90°, is 

1 /¥ 

'' = ¥*• V 7' 

and to the first azimuth of 270°, 

3 Fa 

and the interval from the azimuth of 90° to the next azimuth of 270°, 

equal to the time of one entire oscillation. 

From Equation (348) we have, after substituting for tan<p its 
value in the relation y = a; tan 9, 







a2y2 

/32a;2 - 


adding unity 


to both members, 






^2 a;2 ^. a2 y2 

/32a;2 


also 


from y 


= a;.tan<p, 

a:2 + y2 



= tan2^.^; 



1 + tan2Y/— •^; 



= 1 + tan2 (p ; 

dividing the last equation by this one, and replacing x^ -j- y^ by its 
value a2 — z^^ from the equation of the surface, we get 

l + tan2W^.i 



1 + tan2 9 ' 

but, neglecting the term involving ^*, 

a2 — s2 = a2 ^2 J 

substituting this above, replacing tan2(p by its value in Equation 
(348), and ^2 by its value in Equation (343), after making 



cos 



2\/i-' = "°^'\/f •'-^'"'V?-'' 



MECHANICS OF SOLIDS 
and reducing bv the relation, 



^|.., + sin^.^|.,= l. 



vre have 



235 



+ gi = «-; 



(349) 



which shows that the projection of the path of the body on the 
plane xy, is an ellipse whose centre is on the vertical radius of the 
sphere, and that the line connecting the body with the centre of 
the sphere, describes a conical surface. 

If a = (3, then wili, Equations (343) and (348), 



tl2 =: k2 _ Q2 . 



t\ 



and. Equation (349^ 



,r2 + 7/2 =r a2 «2 ^35Qj 

hence, the body will describe a horizontal circle with a uniform 
motion. 

The pressure upon the surface, at any point of the body's path, 

is given by the value of N in Equation (334). 

§223. — Example 2. — Let the body, still reduced to its centre of 
inertia and acted upon by its 
own weight, be also repelled 
from the bottom point A 
of the bowl, by a force w^hich 
varies inversely as the square 
of the distance ; required the 
position of the body in which 
it would remain at rest. 

As the body is to be at 
rest, there will be no inertia 
exerted, and we have 











cC^y 






; 



236 ELEMENTS OF ANALYTICAL MECHANICS. 

and assuming the axis z vertical, positive upwards, and the origin 
at the lowest point A^ 

L = x'' + y"^ ^ z^ -2az :=^0, • . . . (351) 

dL ^ dL ^^ dL 

— = 2^; ^ = ^2/; ^ = 2 (. - a) ; 

and denoting the distance of the body from the lowest point by r, 
the intensity of the repelling force at the unit's distance by F^ and 
the force at any distance by P, then will 

P=— ; r ::=. V^2 _|_ yZ 4_ ^2. . . . . (352) 

X ^ y z 

for the force P, cos a = — ; cos p = — ; cos 7 =: — : for the 

w^eight Mg^ cos a' = 0; cos ^' = ; cos 7' = — 1 ; and 

Fx Fv Fz 

^=^; ^^ = -/; z=-M<,+ -f. 

These several values being substituted in Equations (319), give 
Fyx Fyx 

The first equation establishes no relation between x and ?/, since 
the equilibrium which depends upon the distance of the particle 
from the source of repulsion, would obviously exisi at any point 
of a horizontal circle whose circumference is at the proper height 
from the bottom. 

From the second equation we deduce, 

Fa ,^ 
-3- = ^^9. 



1 

F_a\\ 
Mg- 

M g ~ a 



(353) 



MECHANICS OF SOLIDS. 237 

from which r becomes known ; and to determine the position of the 
circle upon which the body must be placed, we have, bj making 
ar — in Equations (352) and (351), 

-y/z^ + y2 = r, 

y2 4- s2 _ 2 a 2 = 0. 

Equation (353) makes known the relation between the weight 
of the body and the repulsive force at the unit's distance ; the in- 
tensity of the force at any other distance may therefore be deter- 
mined. 

If there be substituted a repulsive force of different intensity, 
but whose law of variation is the same, we should have, in like 
manner, 

Mg ~ a ' 
hence, 

F'.F'.: r3 :r'3; 

that is, the forces are as the cubes of the distances at which the 
body is brought to rest. 

If, instead of being supported on the surface of a sphere, the 
body had been connected by a perfectly light and inflexible line 
with the centre of the sphere and the surface removed, the result 
would have been the same. In this form of the proposition, we 
have the common Electroscope. 

The differential co-eflicients of the second order, or the terms which 
measure the force of inertia, being equal to zero, Equations (332), 
show that the resultant of the extraneous forces, in this case the 
weight and repulsion, is normal to the surface, which should be the 
case ; for then there is no reason why the body should move in 
one direction rather than another. The pressure upon the surface is 
given by the value of iV, in Equation (334). 

g 224. — Example 3. Let it be required to find the circumstances 



238 



ELEMENTS OF ANALYTICAL MECHANICS. 



of motion of a body acted upon by its own weight while on the 
arc of a cycloid, of which 
the plane is vertical, and 
directrix horizontal. 

Taking the axis of z, 
vertical; the plane zx^ in 
the plane of the curve; 
and the origin at the low- 
est point, then will 




s/Z 



—1 z 
z ~ z^ — a versin — =0 ; 
a 



in which z is taken positive upwards. 



dL _ dL 

dx ~ ' dz 



/2a — z 
and Equation (330) becomes 



df^ 



■sj^ 



— ^ , , d'^z 



and by transposition and division, 



d^ 
df^ 



c^2, 



/2 a — 2 dt^ i'^a — z 



\/^- "V- 

From the equation of the curve we find, 



"Idx — 2dz 



'2 a — z 



multiplying by Equation (357), there will result 



2dx.d^z 
dt^ 



2ffdz~ 



2dz.d^z 
d~t^ 



(354) 



(355) 



(356) 



(357) 



(358) 



MECHANICS OF SOLIDS. 239 

and by integration, 



= F2= C-2gz', 



or, 

and supposing the velocity zero, when = A ; 

which subtracted from the above gives 
dx^ -{- dz^ 



di^ 



2g(h-z)', (359) 



and eliminating dx"^ by means of Equation (358), 



^=!-('--) 



whence, 

dt^—J- 
V 9 



■^h z — z^ 



the negative sign being taken because 2 is a decreasing function 
of U 

By integration, 

fa p dz foT . -I 2z , ^ 

y 9 J ^hz - 22 V ^ A 

Making 2 =: ^, we have 

/a — 1 

r= — V — • versin 2 + (7 ; 



240 ELEMENTS OF ANALYTICAL MECHANICS, 

whence. 



C = tf 



and 



^ = y^^(^ - versin ^ ^) • .... (360) 
When the body has reached the bottom, then will 2 = 0, and 



/ o. 
V 9 

which is wholly independent of h, or the point of departure, and 
we hence infer that the time of descent to the lowest point wdll be 
the same in the same cycloid, no matter from what point the body 
starts. 

Whenever z z= h, the body will, Equation (359), stop, and we 
shall have the times arranged in order before and after the epoch, 

the difference between any two consecutive values being 

"ItiK — 

V y 

The body will, therefore, oscillate back and forth, in equal times. 
The cycloid is, on this account, called a Tautochronous curve. 

The pressure upon the curve is given by Equation (334). 

The time being given and substituted in Equation (360), the value 
of z becomes known, and this, in Equations (359) and (358), will 
give the body's velocity and place. 

§225. — Example 4. — Let a body reduced to its centre of inertia, 
and whose weight is denoted by W^ be supported by the action 
of a constant force upon the branch E H oi an hyperbola, of which 
the transverse axis is vertical, the force being directed to the centre 
of the curve. Required the position of equilibrium. 



MECHANICS OF SOLIDS. 



241 



Denote the constant force by W\ which may be a weight at the 
end of a cord passing over a small wheel 
at C, and attached to the body M. De- 
note the distance CM by r, and the axes 
of the curve by A and B. Take the axis 
z vertical, and the curve in the plane xz. 
Make 



P" ^ W 



then will 



cos 



cos 7' = 1, cos a' = 0, 

Z X 

''=-—, cos a" = - ~, 
r r 




X = P' cos a' + P" cos a" = 



W 



Z =: P' cos 7' 4- P" COS v" = TT - TF'. — , 



and as the question relates to the state of rest, 

0. 






The Equation of tho curve is 

L = A^x^ -B^z^ + A^B^ = 0; 



whence, 



dL 

dx 

dL 

dz 



= 2A^x, 
= -2B^z: 



these values substituted in Equation (330), give 



whence, 



W'B^ — - WA^x + W'A^ — 
r r 



16 



(301) 



242 ELEMENTS OF ANALYTICAL MECHANICS. 

But 

r2 = a;2 + 2;2 ^ s2 + —z^ - B^ = z^ ^^ ^2; 

whence denoting the eccentricity by e, 

r = ■yje'z' - ^2 

and this, in Equation (361), gives after reduction, 

_ B . W 

^ - e(W^-W'^ e2)2 ' 

which, with the equation of the curve, will give the position of 
equilibrium. 

If W e be greater than W, the equilibrium will be impossible* 
]f W e = W, the body will be supported upon the asymptote. 

The pressure upon the curve is given by Equation (334). 

§ 226. — Example 5. — Required the circumstances of motion of a 
body moving from rest under the action of its own weight upon an 
inclined right line. 

Take the axis of z vertical, 
the plane z x to contain the 
line, and the origin at the 
point of departure, and let z 
be reckoned positive down- 
wards. Then will 

L=:z~ax = 0, 

dj^ _ . dL _ 
dz ~ ' dx ~ "'"' 

which in Equation (330) give, after omitting the common factor M, 




d^x , d-'z 



(362) 



From the equation of the line we have 

d^x z= 



MECHANICS OF SOLIDS. 24:3 



which in Equation (362), after slight reduction, 



Multiplying by 2dz^ and integrating, 

the constant of integration being zero. 
Whence 



and 



,^ /2(1 +a2) dz 

J^M^, .^^11+^).,. . . . (303) 
y go? y gaz-z 



the constant of integration being again zero. 

The body being supposed at B^ then will z ^:i AD\ and if we 
draw from B the perpendicular B C to AB^ we have 



AB^ l + a' 



which substituted above, 



/aB" 2 f^. 

='V-7-\g=\/-T' 



(364) 



in which c? denotes the distance A C. 

But the second member is the time of falling freely through the 
vertical distance d] if, therefore, a circle be described upon A C sls 
a diameter, we see that the time down any one of its chords, ter- 
minating at the upper or lower point of this diameter, will be the 
same as that through the vertical diameter itself. This is called the 
mechanical property of the circle. 

Example 6. — A spherical body placed on a plane inclined to the 
horizon, would, in the absence of friction, slide under the action of 
its own weight; but, owing to friction, it will roll. Required the 
circumstances of the motion. 



244 



ELEMENTS OF ANALYTICAL MECHANICS. 



If the sphere move from rest with no initial impulse, the centre 
■will describe a straight line 
parallel to the element of 
steepest descent. Take the 
plane xz, to contain this 
element, the axis z vertical 
and positive upwards. 

The equation of the path 
will be, 




L z= z -\- X tan a — h 



whence, 



dz 



dL 

dx 



= tana. 



The extraneous forces are the weight of the sphere and the fric- 
tion. Denote the first by PF, and the second by F. The nature 
of friction and its mode of action will be explained in the proper 
place, § 307 ; it will be sufficient here to say that for the same 
weight of the sphere and inclination of the plane, it will be a con- 
stant force acting up the plane and opposed to the motion. We 
shall therefore have 



Z — — Mg -\- F^m a ; X = — i^cosa, 
which values, and those above substituted in Equation (330), give 



— • F cos a — M 



d^x 



^^2 + {^9 - ^sina + ^'jj) tana = 0. 

But from the equation of the path, we have 
d'^z = — c?2 a; • tan a ; 
and eliminating d'^x by means of this relation, there will result 
d?'z . (F , \ 



MECHANICS OF SOLIDS. 245 

Multiplying by 2dz^ integrating and making the velocity zero 
when z = h, v^e have 

d z^ /F \ 

jj=V,' = 2sina {^ -gsina) . (z - h). 

This gives 



W2sina(— — ^sina) ^' 



and by integration, the time being zero when z ~ h^ 

F 

A — 2 = -J sin a (^ • sin a 5? ) * ^^' • • * (^)' 

Again, all axes in the sphere through its centre, are principal 
axes ; the sphere will only rotate about the movable axis y, in 
which case v^ and v, will each be zero, and Equations (228) will give 



wherein, 



^=^^'^ 4^-S^ ^'=^-' 



r being the radius of the sphere. 
Whence, 

<f2+ Fr 



Multiplying by 2d-]j, integrating, and making the angular velocity! 
and the arc v)^ vanish together, 

di^ ~ Mk}"^' 
whence, 



[MkJ 



/+ 



246 ELEMENTS OF ANALYTICAL MECHANICS, 

and by integration, making t and 4^ vanish together, 

F r 

Also, because the length of path described in the direction of the 
plane is r.-^^, we have, in addition, 

A — s =1 r . 4/ . sin a ; 

and eliminating 4^ from this and the above equation, there will 
result 



< = x //f^;' (h-z). («) 

Dividing Equation (a) by Equation (6), and solving with respect 
to F, 

^=^ "''''' kj^^'' ^^^ 

and this in Equation [b), gives 



V ^ • sin^ a H ^ ' 

If the sphere be homogeneous, then will 



72 2 2 A ; /2(h-z) /7 

if the matter be all concentrated into the surface, then will 

' ^ V ^ . sui2 a V 3 

which times are to one another as -Y/2r to -y/^K 



CONSTRAINED MOTION ABOUT A FIXED POINT. 

§ 227.— If a body be retained by a Jixed point, the fixed and 
what has been thus far regarded as a movable origin may both be 
taken at this point; in which case, dxj, d 9/^, Sz^, in Equation (40), 
•will be zero, the first three terms of that general equation of equi- 



MECHANICS OF SOLIDS. 247 

librium will reduce to zero independently of the forces, and the equi- 
librium will be satisfied by simply making 

2 P (a; cos jS — y cos a) — 2 m 



2 P (2 cos a — a; cos 7) —2m 



2 P (y cos y — z cos ^) — 2 m > 







df' 




V 


z. 


dr- 


X — xd'^z 


= 






df^ 




y_ 


.d'^ 


z — z 


C?2y 


= 



(365) 



the accents being omitted because the elements m, m\ &c., being 
referred to the same origin, x\ y\ z' will become ar, y, 0. 

The motion of the body about the fixed point might be discussed 
both for the cases of incessant and of impulsive forces, but the discus- 
sion being in all respects similar to that relating to the motion about 
the centre of inertia, § 169 and § 187, we pass to 

CONSTEAmED MOTION- ABOUT A FIXED AXIS. 

§228. — If the body be constrained to turn about a fixed axis, 
both origins may be taken upon, and the co-ordinate axis y to 
coincide with this axis; in which case ^a;^, ^y,-, <^2^, <^9 and ^^, 
in Equation (40), will be zero, and to satisfy the conditions of 
equilibrium, it will only be necessary for the forces to fulfil the 
condition, 

2 P (2 cos a — a; COS7) —2m ^r^~ ~ = • • (366) 

the accents being omitted for reasons just stated. 

§229. — The only possible motion being that of rotation, let us 
transform the above equation so as to contain angular co-ordinates. 
For this purpose we have. Equations (36), 

a:' = r"sin^; 2' = /' cos 4. (367) 

in which r" denotes the distance of the element m from the axis y. 
Omitting the accents, differentiating and dividing by dt^ we have 

dx d\ dz d\ 

^ = rcos + — ; _ = _.sm+.-^. • • (368) 



248 ELEMENTS OF AIJ'ALTTICAL MECHANICS. 

. Now, 

Z'd'^x X'd?z _ 1 / ^_ ^\ 
"T^^ dW - Ti V' dt ""dtJ'^ 

whence by substitution, Equations (367) and (368), 

and since — % must be the same for every element, we have, Equa- 
tion (366), 

72 1 

2 m r^ . __Z = 2 P (^ cos a, — x cos r), 
df- ^ ' ' 

and 

c?2+ _ 2P.(gcosa — rrcos/) 

"^ - 2^;^7^ .... (dbJ) 

That is to say, the angular acceleration of a body retained by a 
fixed axis, and acted upon by incessant forces, is equal to the 
moment of the impressed forces divided by the moment of inertia 
with reference to this axis. 

Denoting the angular velocity by V^ , and the moment of inertia 
by /, we find, by multiplying Equation (369) by 2 c? 4^ and integrating, 

/ Fi2 = 2y2 P (2 cos a ~ a; cos/) d^ -^ C, 

and supposing the initial angular velocity to be Fj', we have 

/(F^2_ Fi'2) = 2 y*2P(0 cos a - a; cos 7) cf 4.. 

But the second member is, § 105, twice the quantity of work 
about the fixed axis ; whence the quantity of work performed be- 
tween the two instants at which the body has any two angular 
velocities, is equal to half the difference of the squares of these 
velocities into the moment of inertia, or to half the living force 
gained or lost in the interval. 



MECHANICS OF SOLIDS. 



249 



If Fi2 — Fi'2 =z 1, we find the value of / to be twice the 
quantity of work required to produce a change in the square of the 
angular velocity equal to unity. 



COMPOUND PENDULUM. 

§ 230. — Any body suspended from a horizontal axis A B, about 
which it may swing with freedom under the 
action of its own weight, is called a compound 
pendulum. 

The elements of the pendulum being acted 
upon only by their own weights, we have 

F = mg; P' — m' g, &c. ; 

the axis of z being taken vertical and positive 
downwards, 

cos a = cos a' = &;c. = ; 

cos y z= cos y' = &c. = 1, 

and Equation (369) becomes 

2 mx 




d^ 
~dF 



'LTTir^ 



(370) 



Denote by e, the distance A G^ of the centre of gravity from the 
axis; by 4-, the angle HAG^ which 
A G makes with the plane yz\ by x^^ 
the distance of the centre of gravity 
from this plane ; then will 

ar^ = e . sin 4/ ; 

and from the principles of the centre 
of gravity, 

2 m a; = Mx^ = M. e . sin 4 ; 

which substituted above, gives 

M. e . sin 4 




dC^ 



— 9 



^mr' 



(371) 



250 ELEMENTS OF ANALYTICAL MECHANICS. 
Multiplying by 2 c? 4', and integrating, 

c/4.2 M.e, . ^ ^ 

-JI2 = ^9'-^ 5-COS4. + C. 

Denoting the initial value of 4^ by a, we have 

Me 
^ = ^9'^ r- cos a +(7; 



whence, 



but 



c?-X2 ^ M.e , , 



<=°^" = ^ -0 + 1:2:3:4 -*'''• 

and taking the value of -4, so small that its fourth power may be 
neglected in comparison with radius, we have 

a2 _ 4.2 
cos 4^ — COS a = -— ^— ; 

which substituted above, gives, after a slight reduction, and replacing 
2 m 7-2 by its value given in Equation (244), 

df= -x/-^-^~ 



v^ 



the negative sign being taken because 4^ is a decreasing function of 
the time. 

Integrating, we have 



t = J^l±Z. cos-' -t (373) 

V e.g a ^ ' 

The constant of integration is zero, because when 4^ = a, we have 
« = 0. 



MECHANICS OF SOLIDS. 251 

Making 4^ = — ^i ^® have 



which gives the time of one entire oscillation, and from which we 
conclude that the oscillations of the same pendulum will be igochro- 
nal, no matter what the lengths of the arcs of vibration, provided 
they be small. 

If the number of oscillations performed in a given interval, say 
ten or twenty minutes, be counted, the duration of a single oscillation 
will be found by dividing the whole interval by this number. 

Thus, let & denote the time of observation, and li the number of 
oscillations, then will 



'V" 



l^ ,^,' + e\ 



JfT- V e.ff ' 

and if the same pendulum be made to oscillate at some other location 
during the same interval &, the force of gravity being different, the 
number JV' of oscillations will be different ; but we shall have, as 
before, ^' being the new force of gravity, 



Squaring and dividing the first by the second, we find 

^-'j <->• 

that is to say, the intensities of the force of gravity, at different 
places, are to each other as the squares of the number of oscilla- 
tions performed in the same time, by the same pendulum. Hence, 
if the intensity of gravity at one station be known, it will be easy 
to fmd it at others. 

§ 231.— From Equation (372), we have 

-^ .2mr2 = 2if.y.e(cos^). - cosa); . . (375) 



252 ELEMENTS OF AI7ALYTICAL MECHANICS, 

and making 

-j^ = ^ij Hmr^ = /; e(cos4/ — cos a) = H\ 

we have 

7. V,^ n: ^M.g,H', (376) 

in which H^ denotes the vertical height passed over by the centre 
of gravity, and from which it appears that the pendulum will come 
to rest whenever 4^ becomes equal to a, on either side of the ver- 
tical plane through the axis. 

§ 232. — If the whole mass of the pendulum be conceived to be 
concentrated into a single point, the centre of gravity must go 
there also, and if this point be connected with the axis by a medium 
without weight, we have what is called a simple pendulum. Deno- 
ting the distance of the point of concentration from the axis by /, 
we have 

A:^ = 0; e = If 
which reduces Equation (374) to 

' = ^-\/j (377) 

If the point be so chosen that 



or, 

I — - ; 



(378) 



the simple and compound pendulum will perform their oscillations in 
the same time. The former is then called the equivalent simple pen- 
dulum; and the point of the compound pendulum into which the 
mass may be concentrated to satisfy this condition of equal duration, 
is called the centre of oscillation. A line through the centre of 
oscillation and parallel to the axis of suspension, is called an axis of 
oscillation. 



MECHANICS OF SOLIDS. 253 

g 233. — The axes of oscillation and of suspension are reciprocal. 
Denote the. length of the equivalent simple pendulum when the com- 
pound pendulum is inverted and suspended from its axis of oscillation, 
by V , and the distance of this latter axis from the centre of gravity 
by e[ then will 

I z=z e -{- e' or e' z= I — e; 
and, Equation (378), 

_ k,'' + e"^ _ k;^ + (^ - eY 
^ - e' - l-^e 

and replacing ^, by its value in Equation (378), we find 

e 

That is, if the old axis of oscillation be taken as a new axis of sus- 
pension, the old axis of suspension becomes the new axis of oscilla- 
tion. This furnishes an easy method for finding the length of an 
equivalent simple pendulum. 

Diflferentiating Equation (378), regarding I and e as variable, we 
have 

dl _ e2 - k^^ 
de ~ e^ 



and if / be a minimum. 



whence, 



de ~ ~ e^ 



ezz:k, 



But when Z is a minimum, then will t be a minimum, Equa- 
tion (377). That is to say, the time of oscillation will be a 
minimum when the axis of suspension passes through the principal 
centre of gyration^ and the time will be longer in proportion as the 
axis recedes from that centre. 



254 



ELEMENTS OF ANALYTICAL MECHANICS. 




Let A and A' he two acute parallel prismatic axes firmly con- 
nected with the pendulum, the acute edges 
being turned towards each other. The 
oscillation may be made to take place 
about either axis by simply inverting the 
pendulum. Also, let M be a sliding mass 
capable of being retained in any position 
by the clamp-screw IT. For any assumed 
position of M, let the principal radius of 
gyration be G C; with G &s & centre, 
G C SiS radius, describe the circumference 
CSS'. From what has been explained, 
the time of oscillation about either axis 
will be shortened as it approaches, and 

lengthened as it recedes from this circumference, being a minimum, 
or least possible, when on it. By moving the mass M, the centre 
of gravity, and therefore the gyratory circle of which it is the 
centre, may be thrown towards either axis. The pendulum bob being 
made heavy, the centre of gravity may be brought so near one of 
the axes, say A\ as to place the latter within the gyratory cir- 
cumference, keeping the centre of this circumference between the 
axes, as indicated in the figure. In this position, it is obvious that 
any motion in the mass M would at the same time either shorten 
or lengthen the duration of the oscillation about both axes, but 
unequally, in consequence of their unequal distances from the gyratory 
circumference. 

The pendulum thus arranged, is made to vibrate about each axis 
in succession during equal intervals, say an hour or a day, and the 
number of oscillations carefully noted; if these numbers be the 
same, the distance between the axes is the length I, of the equiva- 
lent simple pendulum ; if not, then the weight M must be moved 
towards that axis whose number is the least, and the trial repeated 
till the numbers are made equal. The distance between the axes 
may be measured by a scale of equal parts. 

§ 234. — From this value of I, we may easily find that of the simple 
second^ s pendulum; that is to say, the simple pendulum which will 



MECHANICS OF SOLIDS. 255 

perform its vibration in one second. Let N^ be the number of 
vibrations performed in one hour by the compound pendulum whose 
equivalent simple pendulum is l\ the number performed in the 
same time by the second's pendulum, whose length we will denote 
by Z', is of course 3600, being the number of seconds in 1 hour, 
and hence. 





I 
J 9 


3600^ = ^ =^\ 


rv 

' 9 



and because the force of gravity at the same station is constant, 
we find, after squaring and dividing the second equation by the first, 

Such is, in outline, the beautiful process by which Kater determined 
the length of the simple second's pendulum at the Tower of London 
to be 39,13908 inches, or 3,26159 feet. 

As the force of gravity at the same place is not supposed to 
change its intensity, this length of the simple second's pendulum 
must remain forever invariable ; and, on this account, the English 
have adopted it as the basis of their system of weights and measures. 
For this purpose, it was simply necessary to say that the 3-,2^t5-9''* 
part of the simple second's pendulum at the Tower of London shall 
be one English foot^ and all linear dimensions at once result from 
the relation they bear to the foot ; that the gallon shall contain 
AW^ of a cubic foot, and all measures of volume are fixed by the 
relations which other volumes bear to the gallon ; and finally, that 
a cubic foot of distilled water at the temperature of sixty degrees 
Fahr. shall weigh one thousand ounces^ and all weights are fixed by 
the relation they bear to the ounce. 

§235. — It is now easy to find the apparent force of gravity at 
London ; that is to say, the force of gravity as affected by the cen- 
trifugal force and the oblateness of the earth. The time of oscillation 



256 ELEMEN"TS OF ANALYTICAL MECHANICS. 

being one second, and the length of the simple pendulum 3,26159 
feet, Equation (377) gives 



= *V^ 



, ^,26159 



whence, 

^ = cr2 (3,26159) = (3,1416)2 . (3,26159) = 32,1908 feet. 
From Equation (377), we also find, by making t one second, 



and assuming 



we have 



? = a; -f y cos 2 4/, 



-^ = :r 4- 2/ cos 2 4^ (380) 

Now starting with the value for g at London, and causing the 
same pendulum to vibrate at places whose latitudes are known, we 
obtain, from the relation given in Equation (374)', the corresponding 
values of g, or the force of gravity at these places ; and these 
values and the corresponding latitudes being substituted successively 
in Equation (380), give a series of Equations involving but two un- 
known quantities, which may easily be found by the method of 
least squares. 

In this way it has been ascertained that 

if^.x = 32,1808 and ir^.y z= — 0,0821 ; 

whence, generally, 

ff = 32,1808 - 0,0821 cos 2 ^]. ; .... (381) 

and substituting this value in Equation (377), and making ^ = 1, 
we find 

/ 
/ = 3,26058 - 0,008318 cos 2 4. • . . . (382) 

Such is the length of the simple second's pendulum at any place 
of which the latitude is %)/. 



MECHANICS OF SOLIDS, 



257 



If we make 4. = 40° 42' 40", the latitude of the City Hall of 
New York, we shall find 



I z=z 3,25938 



39,11256. 



§23C. — The principles which have just been explained, enable us 
to find the moment of inertia of any bodj turning about a fixed 
axis, with great accuracy, no matter what its figure, density, or the 
distribution of its matter. If the axis do not pass through its centre 
of gravity, the body will, when deflected from its position of equi- 
librium, oscillate, and become, in fact, a compound pendulum ; and 
denoting the length of its equivalent simple pendulum by /, we have, 
after multiplying Equation (378) by if, 

M.Le = M (^,2 + e2) _ 2 mr2 ; . . . . (383) 
or since 



W 

M= ~y 
9 

W 

— • l.e = :Smr^, 

9 



(384) 



in which W denotes the weight of the body. 

Knowing the latitude of the place, the length V of the simple 
second's pendulum is known from Equation (382) ; and counting the 
number iV^ of oscillations performed by the body in one hour 
Equation (379) gives 

V • (3600)2 



I =: 



^72 



To find the value of e, which is 
the distance of the centre of gravity 
from the axis, attach a spring or 
other balance to any point of the 
body, say its lower end, and bring 
the centre of gravity to a horizontal 
plane through the axis, which posi- 
tion will be indicated by the max- 
imum reading of the balance. De- 
noting by a, the distance from the axis C to tlic point of support i2, 

17 




258 ELEMEN'TS OF ANALYTICAL MECHANICS. 

and by 5, the maximum indication of the balance, we have, from 
the principle of moments, 

ha = We. 

The distance a, may be measured by a scale of equal parts. Sub- 
stituting the values of TF", e and I in the expression for the moment 
of inertia. Equation (384), we get 

'-^ij^ = I. (385) 

If the axis pass through the centre of gravity, as, for example, 
in the fly-wheel^ it will not oscillate; in which case, take Equation 
(383), from which we have 

Mk;' =z M.l.e - Me^. 

Mount the body upon a parallel axis A^ not passing through the cen- 
tre of gravity, and cause it to vibrate 
for an hour as before; from the num- 
ber of these vibrations and the length 
of the simple second's pendulum, the 
value of I may found; M is Icnown, 
being the weight W divided by g ; and 
e may be found by direct measure- 
ment, or by the aid of the spring 
balance, as already indicated; whence k^ becomes known. 



MOTION OF A BODY ABOUT AN AXIS UNDER THE ACTION OF IM?UL- 

SIVE FORCES. 

I 237. — If the forces be impulsive, we may, § 184, replace in 
Equation (366) the second differential co-efficients of rr, y, 0, by the 
first differential co-efficients of the same variables, which will reduce 
it to 

^ , . zdx — xdz 
iPi^z cos a — xcQsy) — liTn,' = ; 




MECHANICS OF SOLIDS. 



259 



and replacing dx and c?2, by their values in Equations (368), we 
find 

d-\> 2 P (^ cos a — re cos 7) 

dt ~ 2 m r2 



(386) 



That is, the angular velocity of a body retained by a fixed ax'is^ and 
subjected to the simultaneous action of impulsive forces^ is equal to the 
sum of the moments of the impressed forces divided by the moment of 
inertia loith reference to this axis. 

BAUSTIO PENDULUM. 

§ 238. — In artillery, the initial velocity of projectiles is ascertained 
by means of the balistic pendulum, 
which consists of a mass of matter 
suspended from a horizontal axis 
in the shape of a knife-edge, after 
the manner of the compound pen- 
dulum. The bob is either made 
of some unelastic substance, as 
wood, or of metal provided with 
a large cavity filled with some 
soft matter, as dirt, which re- 
ceives the projectile and retains 
the shape impressed upon it by the 
blow 

Denote by V and m, the initial velocity and mass of the ball ; 
Fj the angular velocity of the balistic pendulum the instant after 
the blow, / and M its moment of inertia and mass. Also let r 
represent the distance of the centre of oscillation of the pendulum 
from the axis A. That no motion may be lost by the resistance 
of the axis arising from a shock, the ball must be received in the 
direction of a line passing through this centre and perpendicular to 
the line A 0. This condition bcintr satisfied, we have 




2 F [z cos a — a: cos y) = r .m. F ; 
2 m r^ = m r^ -f- /; 



260 ELEMENTS OF ANALYTICAL MECHANICS. 

and Equation (386) becomes 

rm V 



r^ + I' 
from which we find 



(387) 



the velocity V, becomes known, therefore, when Fj is known, since 
all the other quantities may be easily found by the methods already 
explained. To find Fi, denote by H, the greatest height to which 
the centre of gravity of the pendulum is elevated by virtue of 
this angular velocity ; then, since the moment of inertia of the ball 
is mr^, § 181, we have, from the principle of the living force. Equa- 
tion (376), 

(/+ mr2) Fi2 = 2(ilf 4- w)^^; 
whence, 

tL±!!Lr!)ZL^2^. 

Denoting by T the time of a single oscillation of the pendulum 
after it receives the ball, we have, by multiplying both terms of 
the fraction under the radical sign in Equation (374) by Jf + m, 
and reducing by the relation, {M + m) {k/^ -f e^) = (i/ + m)F, 
Equation (244), 






{M + m)D.g 

D being the distance from the axis to the centre of gravity ; whence, 

I ^ mr'^ _ D ^2^ 
(M + m) g ~ ^2 ' 

and this value, substituted in the equation of the living force, gives 
-^F,2 = 2Zr: 

•^2 

whence, 



MECHANICS OF SOLIDS. 



261 



also, 



1 + mr^ = — ; 



and because, Equation (377), 



T = -r 



we find 



T-'g 



Substituting these values of Fj, 7 + wi r"^ and r in Equation (387), 

we find 

M •\- m 



F = -v5iro. 



m 



or, replacing the masses by their values in terms of the weights 
and force of gravity, 

T w 

in which W and w denote the weights of the pendulum and ball 
respectively. 

Observe that H^ is the height to which the centre of gravity 
rises in describing the arc of a circle of 
which D is the radius. Let G G' K be 
half of the circumference of which this arc 
is a part, G and G' the initial and termi- 
nal positions of the centre of gravity du- 
ring the ascent ; draw G' R perpendicular 
to K G. Tlicn, because A G — D^ and 
G R z=i 11^ wc have, from the property 
of the circle, 




RG' = ^IL{'ID - 11) ; 

and if the pendulum be made large, so that the arc G G' shall be 
very small, which is usually the case, // may be neglected in com- 
parison with 2i>, and therefore 

RG' ^ ^/2H.l) ; 



262 ELEMENTS OF ANALYTICAL MECHANICS. 

-y/^H D is half the chord of the arc described by the centre of 
gravity in one entire oscillation. Denoting this chord by (7, and 
substituting above, we have 

From this equation, we may find the initial velocity V ; and 
for this purpose, it will only be necessary to have the duration 
of a single oscillation, and the amplitude of the arc described by 
the centre of gravity of the pendulum. The process for finding 
the time has been explained. To find the arc, it will be suffi- 
cient to attach to the lower extremity of the pendulum a pointer, 
and to fix on a permanent stand below, a circular graduated groove, 
whose centre of curvature is at A\ the groove being filled with 
some soft substance, as tallow, the pointer will mark on it the 
extent of the oscillation. Knowing thus the arc, denoted by ^, and 
the value of i>, found as already described, § 236, we have 



whence, 
and finally, 



C = 2i).sini 



2 



r = — »D» ^ sini^. (388) 



PART II. 



MECHAIICS OF FLUIDS 



H^TRODUCTORY REMARKS. 

§289. — The physical condition of every body depends upon the 
relation subsisting among its molecular forces. When the attrac- 
tions prevail greatly over the repulsions, the particles are held firmly 
together, and the body is solid. In proportion as the difference be- 
tween these two sets of forces becomes less, the body is softer, and 
its figure yields more readily to external pressure. When these 
forces are equal, the particles will yield to the slightest force, the 
body will, under the action of its own weight, and the resistance 
of the sides of a vessel into which it is placed, readily take the 
figure of the latter, and is liquid. Finally, when the repulsive ex- 
ceed the attractive forces, the elements of the body tend to separate 
from each other, and require either the application of some extra- 
neous force or to be confined in a closed vessel to keep them 
together ; the body is then a gas. In the vast range of relation 
among the molecular forces, from that which distinguishes a solid to 
that which determines a gas or vapor, bodies are found in all possible 
conditions — solids run imperceptibly into liquids, and liquids into 
gases. Hence all classification of bodies founded on their physical 
properties alone, must, of necessity, be arbitrary. 

§240. — Any body whose elementary particles admit of motion 



264 ELEMENTS OF ANALYTICAL MECHANICS. 

among each other, is called a Jluid — such as water, wine, mercury, 
the air, and, in general, liquids and gases ; all of which are distin- 
guished from solids by the great mobility of their particles among 
themselves. This distinguishing property exists in different degrees 
ill different liquids — it is greatest in the ethers and alcohol ; it is 
less in water and wine ; it is still less in the oils, the sirups, 
greases, and melted metals, that flow with difhculty, and rope -when 
poured into the *air. Such fluids are said to be viscous, or to possess 
viscosity. Finally, a body may approach so closely both a solid and 
liquid, as to make it difficult to assign it a place among either 
class, as paste, putty, and the like. 

§241. — Fluids are divided in mechanics into two classes, viz.: 
compressible and incompressible. The term incompressible cannot, in 
strictness of propriety, be applied to any body in nature, all being 
more or less compressible ; but the enormous power required to 
change, in any sensible degree, the volumes of liquids, seems to 
justify the term, when applied to them in a restricted sense. The 
gases are highly compressible. All liquids w^ill, therefore, be regarded 
as incompressible ; the gases as compressible. 

§ 242. — The most important and remarkable of the gaseous bodies 
is the atmosphere. It envelops the entire earth, reaches far beyond 
the tops of our highest mountains, and pervades every depth from 
which it is not excluded by the presence of solids or liquids. It 
is even found in the pores of these latter bodies. It plays a most 
important part in all natural phenomena, and is ever at work to 
influence the motions within it. It is essentially composed of oxygen 
and nitrogen, in a state of mechanical mixture. The former is a 
supporter of combustion, and, with the various forms of carbon, is 
one of the principal agents employed in the development of mechan- 
ical power. 

The existence of gases is proved by a multitude of facts. Con- 
tained in an inflexible and impermeable envelope, they resist pressure 
like solid bodies. Gas, in an inverted glass vessel plunged into 
water, will not yield its place to the liquid, unless some avenue of 
escape be provided for it. Tornadoes which uproot trees, overturn 



MECHANICS OF FLUIDS. 265 

houses, and devastate entire districts, are but air in motion. Air 
opposes, by its inertia, the motion of other bodies through it, and 
this opposition is called its resistance. Finally, we know that wind 
is employed as a motor to turn mills and to give motion to ships 
of the largest kind. 

§ 2.43. — In the discussions which are to follow, fluids will be con- 
sidered as without viscosity ; that is to say, the particles will be 
supposed to have the utmost freedom of motion among each other. 
Such fluids are said to be perfect. The results deduced upon the 
hypothesis of perfect fluidity will, of course, require modification 
when applied to fluids possessing sensible viscosity. The nature and 
extent of these modifications can be known only from experiments. 



MARIOTTES LAW. 

§244. — Gases readily contract into smaller volumes when pressed 
externally ; they as readily expand and regain their former dimen- 
sions when the pressure is removed. They are therefore both com- 
pressible and elastic. 

It is found by experiment, that the change in volume is, for a 
constant temperature, always directly proportional to the change of 
.pressure. The density of the same body is inversely proportional to 
the volume it occupies. If, therefore, F denote the pressure upon 
a unit of surface which will produce, at a given temperature, say 
32° Fahr., a density equal to unity, and I) any other density, and 
p the pressure upon a unit of surface which will, at the same tem- 
perature of the gas, produce this density, then, according to the ex- 
periments above referred to, will 

p = F.D (389) 

This law was investigated by Boyle and Mariotte, and is known 
as Mariotte's Law. By experiments made at Paris, it was found that 
this law obtains, when air, in its ordinary condition, is condensed 27 
and. rarefied 112 times. 



266 ELEMENTS OE ANALYTICAL MECHANICS. 
LAW OF THE PKESSUEE, DENSITY, AKD TEMPEEATUEE. 

§ 245. — Under a constant pressure, all bodies are expanded by 

heat ; under a constant volume, their elastic force is increased by the 

same agent. Experiment has shown that the laws of these changes 

for gases are expressed by 

p = P . i) . (1 + a ^) ; (390) 

in which p denotes the pressure upon a unit of surface, I) the 

density of the gas, ^ the difference between the actual and some 

standard temperature, and a a constant which is equal to -^\^ = 0,00208 

when the standard is 32° Fahr., and & is expressed in units of that scale. 

First supposing D and ^ variable and p constant; then p and & 

variable and D constant, Equation (390) gives 

d D a. D dp ap 

'dl^ ~ 1 +a& ' ~d& "^ 1 +a& • • • • (a) 

The quantity of heat, denoted by q, necessary to change the tem- 
perature ^ degrees from the assumed standard, will be a function 
of p, Z), ^ ; but because of Equation (390,) we may write 

q=f{i>,p) (b) 

The increment of heat which Mill raise a body's temperature one 
degree, is called its specific heat. The specific heat being the in- 
crement of q for each unit of ^, if c denote the specific heat when 
the pressure is constant, and c^ that when the density is constant, 

then will 

dq dq dD dq dq dp 

^~d'&^d~D"d^' ^'"^IF^dp'U' 
or, Equations (a), 

d q a . D d q a .p 

^^ ~td' rr^ ' "^'^ jp' FT "^ *' 

and by division, making c =i y .c^, 

in which y, denotes the ratio of the specific heat of the gas at a 
constant pressure to that at a constant density. This ratio is 
known from experiment to be constant for atmospheric air, and is 
probably so for all gases. The experiments of Desormes and 



MECHANICS OF FLUIDB. 267 

Clements make its value 1,3482; those of Gay-Lussac and "Walter 
1,3748; and those of Dulong on perfectly dry air 1,421. Regard- 
ing y as constant, the integration of the foregoing equation gives 

1 

)7^ 



=/(9 



(See Appendix No. 3.) 



in which /, denotes any arbitrary function of the quantity within 
the parenthesis, and from which, denoting the inverse functions by 
F^ we may write 

p=D^.F{q) (c) 

From Equation (390), we have 

Sudden compression increases, and a sudden expansion decreases the 
temperature of bodies, and if q remain the same, while suddenly 
j9, D, ^, become p\ D\ 6', we have 

f=D-'^.F(q), ■ . (e) 6' = -lj,J)'^-'.F(g)~l. • • (g) 

Eliminating F {q) first from Equations (c) and (e), and then from Equa- 
tions (d) and (g), we have, replacing y and a by their numerical values, 

P'=f{-jj) .... (391) 

/^/N 0,3482 

6'= (480 + 6) (jjj - 480 . . . • (392) 

These equations give the rclaiion bclwccn the densities, elastic 
forces, and the teniperatures of a gas suddenly compressed or dila- 
ted, and retaining the quantity of its licat unchanged. 

The pressure being, constant, make in, Eq. (390), d — 0, D = D^, and 
divide same equation by the result ; we find I) =z D^ — (1 -i-ai)). Make 
p=:lJ,„- h^j ' <j' = weight of a column of niereiiry at standard tenij)i'i'atue, 
T, and resting on a base unity, in Lat. 4;")°, where gravity is y'. '1 he>e 
in Eq. (389) give, after writing ().0(;208 for a, ajid r —32^ for (), 

P = —""-l^L^ . [1 -f {1° - 32°) . 0,00208] . . . (393) 
If the temperature of the mereiij-y vary from the standard T, 



208 



ELEMENTS OF ANALYTICAL MECHANICS. 



and become T' then will D^ also vary and become D[,^ and to exert 
the same pressure li^^ must have a new height 7i, and such that 

D„,.h^,.g' =z 'D[,.h.g'. 
Mercury expands or contracts 0,0001001*^ part of its entire vol- 
ume for each degree of Tahr. by which it increases or diminishes 
its temperature. And as the density of the same body varies 
inversely as its volume, we have 

Dl^B„\\-\-{T- T) . 0,0001001] 
which substituted above gives 

h^^ ^h\\ + {T- T). 0,0001001] (394) 

EQUAL TRANSMISSION OF PKESSUEE. 



§ 246. — Let EH L^ represent a closed vessel of any shape, with 
which two piston tubes A B' and 
D C communicate, each tube be- 
ing provided with a piston that 
fits it accurately and which may 
move wdthin it with the utmost 
freedom. The vessel being filled 
with any fluid, let forces P and 
P', be applied, the former per- 
pendicularly to the piston A B^ 
and the latter in like direction 
to the piston (7i), and suppose 
these forces in equilibrio, which 

they may be, since the fluid cannot escape. Now let the piston 
^ ^ be moved to the position A' B' \ the piston CD will take 
some new position, as CD'. And denoting by s and s\ the dis- 
tances A A' and (7(7', respectively, we have, from the principle of 
virtual velocities, 

Ps = P's'. 

Denote the area of the piston ^^ by a, and that of the piston 
CD hj a\ then will the volume of the fluid which was thrust from 
the tube A B\ be measured by a . 5, and that which entered the tut)e 




MECHANICS OF FLUIDS 269 

D C\ will be measured by a' s'. But the pressure upon the pistons 
and the temperature remaining the same, the entire Tolume of the 
fluid in the vessel and tubes will be unchanged. Hence, 

as = a' s' ] 

dividing the equation above by this one, we have 

P P' 

- = ^ (396) 

a a 

That is to say, two forces applied to pistons which communicate freely 
with each other through the intervention of some confined fluids will 
he in equilihrio when their intensities are directly proportional to the 
areas of the pistons upon which they act. 

This result is wholly independent of the relative dimensions and 
positions of the pistons ; and hence we conclude that any pressure 
communicated to one or more elements of a fluid mass in equilibrio, is 
equally transmitted throughout the whole fluid in every direction. This 
law which is fully confirmed by experiment, is known as the prin- 
ciple of equal transmission of pressure. 

§247. — Let a become the superficial unit, say a square inch or 
square foot, then will P be the pressure applied to a unit of sur- 
face, and, Equation (396), 

P' = Pa'. (397) 

That is, the pressure transmitted to any portion of the surface of 
the containing vessel, will be equal to that applied to the unit of 
surface multiplied by the area of the surface to which the transmis- 
sion is made. 

§ 248 — Since the elements of the fluid are supposed in equilibrio, 
the pressure transmitted to the surface through the elements in con- 
tact with it, must, § 217 and Equations (332), be normal to the sur- 
face. That is, the pressure of a fluid against any surface^ acts always 
in the direction of the normal. 



270 



ELEMENTS OF ANALYTICAL MECHANICS. 



MOTION OF THE FLTJED PAUTICLES. 

§ 249. — The particles of a fluid having the utmost freedom of 
motion among one another, all the forces applied at each particle 
must be in equilibrio. Eegarding the general Equation (40) as ap- 
plicable to a single particle, whose co-ordinates are x^ y, 2, we shall 
have 

and supposing the particle to have simply a motion of translation, 
we also have 

5(p = Oj d-\, = 0; d'us z=z 0; 

and that equation becomes 

(d^ x\ 
2 P cos a — m • -y-^ ) ^ ^ 

+ (2Pcos^-m.^)(^y 1^=0; 

+ y^PGOsy —m'-j-^J z 

whence, upon the principle of indeterminate co-efficients, 

d'^x 



2 P cos a 



^'T¥ = ' 



2Peos/3-m.|^ 

d'^z 
2 P cos 7 - wi • -^ 



0. 



(398) 



Now the terms 2 P cos a, ^ P cos /3 and 2 P cos 7, are each composed 
of two distinct parts, viz. : 1st., the component of the resultant of 
the forces applied directly to the particle ; and 2d., the component 
of the pressure transmitted to it from a distance, arising from the 
forces impressed upon other particles. 

Denote by X, Y and Z, the accelerations, in the directions of the 
axes x^ y, 0, respectively, due to the forces applied directly to the 



MECHAJ^ICS OF FLUIDS, 



271 



particle; then m, being the mass of the particle, the components of 
the forces directly impressed will be 

mX ) m Y; w.Z. 

The pressure transmitted will depend upon the particle's place, 
and will be a function of its co-ordinates of position. Denote by ^, 
the pressure upon a unit of surface, on the supposition that every 
point of the unit sustains a pressure equal to that communicated to 
the particle from a distance; then will 

Conceive each particle of the fluid to consist of a small rectan- 
gular parallelopipedon whose 
faces are parallel to the co- 
ordinate planes, and whose con- 
tiguous edges at the time if, 
are dx^ dy and dz\ and let 
x^ y, 0, be the co-ordinates of 
the molecule in the solid an- 
gles nearest the origin of co- 
ordinates. Then would the 
difference of pressure on the 
opposite faces, which are paral- 
lel to the plane zy^ were these faces equal to unity, be 

dp 




F{x-^ dx, y,z,) - F{x,y,z,] 



c/.-^^' 



and upon the actual faces whose dimensions are each dz.dy^ this 
difference becomes, Equation (397), 

-T— • dx'dy ' dz. 

dx -^ 

In like manner will the difference of the pressures transmitted 
to the opposite faces parallel to the planes zx- and xy^ be, respeo- 
tively, 



dp 
dy 



dy ' dz ' dx^ and 



dp 
d z 



dz ' dx 'dy. 



272 



ELEMENTS OF ANALYTICAL MECHANICS. 



These pressures being normal to the surfaces to which they are 
respectively applied, they will act, the first in the direction of x, 
the second in the direction of y, and the third in the direction 
of z. And as these differences alone determine that portion of the 
motion due to the transmitted pressures, we have 



2 P cos a 

2 P cos /3 =mY 
2 P cos y = m Z 



mX ; dx .dy .dz\ 

dx 



dp 
dy 



dz 



' dy ,dx .dz\ 



dz .dx . dy. 



Denote by D the density of the mass m^ then will, Equation (1)', 
m =D.dx.dy.dZj 
and by substitution, E([uations (398) become 



dr 



= X 



d-^x 



D 


dx ~ 


-^^ dt^^ 


1 


dp 
dy - 


dt^ ' 


1 
D 


dp 

dz ~ 


- ^ dt^ ^ 



(399) 



Denote by m, v and w^ the velocities of the molecule whose co- 
ordinates are xyz^ parallel to the axes rr, y, ^, respectively, at the 
time t. Each of these will be a function of the time and the co- 
ordinates of the molecule's place; and, reciprocally, each co-ordinate 
will be a function of ^, w, v and w ; whence. Equations (12) and (13), 



d'^x 
If 



du ^ /'du\ dt du dx du dy du dz 
It ~ \dl/ "d~t d^ "d~t ~dy "d't ~^ Jz "d~t ' 



d X d 1/ d z 
and replacing -n' j^' ~^' ^7 ^^eir values w, v, w;, respectively, we 

have 



d^x 



/du\ du du du 



dy 



dz 



MECHANICS OF FLUIDS. 
in the same way, 

^lLL - /^-^^ -1- 
d t^ ~ \dt / dx 

d'^z /dw\ dw , dw , dw 

c/^2 \dt/dx ' dy ^ dz 

which, substituted in Equations (399), give 



273 



dv , dv dv 

. M + — - . V + —- . w, 
dy dz 



1 


dp 
dx 


= X^ 


m^ 


du du 

. 11 — . 

dx dy 


du 

• V — — — 'W 

dz 


1 

D 


dp 
dy 


= Y- 


m- 


dv dv 
dx dy 


dv 

• V ; W 

dz 


1 


dp 
dz 


= Z - 


(w)^ 


dw dw 

• u 

d X dy 


dw 

V ; W 

dz 



(400) 



Here are three equations involving five unknown quantities, viz. •. 
w, V, w, 2^ ^^^ ^1 which are to be found in terms of x, y, z and t. 

Two other equations may be found from these considerations, viz : 
the velocity in the direction of x, of the molecule whose co-ordinates 
are xyz, is w, the velocity of the molecule in the angle of the 
parallclopipedon at the opposite end of the side d x, at the time /, is 



u + 



du 
dx 



x: 



and hence the relative velocity of the two molecules is 



du _ du _ 

u + - — dx — It z= - — 'dx. 
dx dx 



At the time t, the length of the edge joining these molecules is 
dx, and at the end of the time t -{- d i, this length will be 

7 du . , T / , du . . 

dx -\- - — ' dx . dt z=z d X {\ + -r— ' dt)'. 
dx ^ dx ' ^ 

the second term being the distance by which the molecules in 
question approach toward or recede from one another in the 
time dt, 

18 



i5?4 ELEMENTS OF A^-ALTTICAL MECHANICS. 

In the same way the edges of the parallelopipedon which at the 
time t^ were dy and ds^ become respectively, 

7 ^ ^ » 7 7/1 dv . . 

dy -\-—'dy.dt z= dy (I +—'dt); 

. dw _ , 7 ,, * dw , , 

dz-\ -—'dz.dt =dz{\ ■\- -^—-di\\ 

dz ^ dz ' 

and the volume of the parallelopipedon, which at the time f, was 
dx .dy . dz^ becomes at the time t -\- dt, 

The density, which was i>, at the time t, being a function of xy z 
and i, becomes at the time t -{■ dty 

I) + -T-r'dt +-T— •c?:^ + --7— -c^y + --r— .(^s; 
a i d X dy dz 

which may be put under the form, 

^ , /dD , dD dx dD dy , dD d z\ , 
Va^ arc a^ a?/ dt dz dt^ 

and replacing 

dx dy dz 
dt dt dt ^ 

by their values u, v, w, respectively, 

\a^ arc ay dz ^ 

Multiplying this by the volume above, we have for the mass of the 
parallelopipedon, which was 

D .dx ,dy , dz^ 

at the time t^ the value, 

.dx.dy.dz{.^'^.dt)\.^^.dt)\,^'^.dt) 
at the time t -{■ d t. 



MECHANICS OF FLUIDS. 275 

But these masses must be equal, since the quantity of matter 
is unchanged. Equating them, striking out the common factors, per- 
forming the multiplication, and neglecting the second powers of the 
differentials, we have 

^ { du dv , dw\ dD dD dD dD ^ ,,^,, 

\dx dy dzy dt dx dy dz ^ ' 

This is called the Equation , of continuity of the fluid. It expres- 
ses the relation between the velocity of the molecules and the den- 
sity of the fluid, which are necessarily dependent upon each other. 
This is a fourth equation. 

§250.— If the fluid be compressible, then will the fifth equation 
be given by the relation, 

F{D,p) = 0, (402) 

a^ is illustrated in the particular instance of Mariotte's law, Equa- 
tion (389). The form of the function designated by the letter i^, 
will depend upon the nature of the fluid. 

§251. — If the fluid be incompressible, the total differential of i> 
will be zero, and 

dD dD dD dD ^ ,,^„, 

and consequently, the equation of continuity. Equation (401), becomes, 

d u dv d w ^ , ^ 

1^ + 1^ + ^='' (^«^) 

and wc have for the determination of u^ v, w, D and p, the five 
Equations (400), (403), (404). 

§ 252. — These equations admit of great simplification in the case 
of at incompressible homogeneous fluid when u-dx-\- v .dy + iv.dz^ 
is a perfect differentia]. For if we make 

udx + vc?y + wdz =. dcp^ 



276 ELEMENTS OF ANALYTICAL MECHANICS. 

then from the partial differentials will 

d(p dcp dcp ,.^^x 

dx dy dz ^ ' 

which, in Equation (404), gives for the equation of continuity, 

^'9 C^2(p .6^2(p 

by the integration of which the function 9 may be found. 
Diflferentiating the values of u^ v and w above, we have 

_ d"^ (n . d^cp , c?2 m 

du = — — ; a V = —. — : d w = — — • 
dx dy dz 

Eliminating u^ ^, ^, du^ d v and d w, from Equation (400), by means 
of the values of these quantities above, we have 

f?2 (p d (p d^cp d(p c?2 (7j c? 9 c?2 (p 



1 
JJ 


dp 
dx 


= X - 


1 
d' 


dp 
dy 


= Y - 


1 

D 


dp 

dz 


z= Z - 



dx'dt dx d x^ dy dx.dy dz dx,dz^ 

d^cp dcp d'^cp d(p d^ o dcp d^cp 

dy .d i dx dy . dx d y d y"^ dz dy .dz ^ 

d^ cp dcp d^ cp dcp d'^ cp dcp d^ cp 

dz.dt dx dz.dx dy dz.dy dz d?? 

Multiplying the first by dx., the second by c?y, the third by dz^ ad- 
ding and we find, 

>=x..+ r.,4-...-.?,-i.[(£)V (^pV (fi)l(407) 

From which, by integration, may be found the pressure at any point 
of an incompressible fluid mass in motion, when Equation (406) is 
the equation of continuity. 

§253. — When the excursions of the molecules are small, the 
second powers of the velocities may be neglected, which will reduce 
Equation (407) to 

^-dp =Xdx ^Ydy ^ Zdz - d^> . • (408) 



MECHANICS OF FLUIDS. 277 

§254. — If the condition expressed by Equation (406) be not fuU 
filled, then we must have recourse to Equation (404) to find the 
pressure. 

§255. — Resuming Equation (401), which appertains to a compres- 
sible fluid, retaining the condition that 

udx + vdy + wdz = dc^ 

is a perfect differential, and from which, therefore, 

„ = i^; . = i^., »=4^; . . . (409) 

dx dy dz ^ ' 

we obtain by substitution, 

^ ( du dv dw ) dD dD do dD dcp dD dcp 

( dx dy dz ) dt dx dx dy dy dz dz 

If the excursions of the molecules from their places of rest be 
very small, both the change of density and velocity will be so 
small that the products which constitute the last three terms of 
t?iis equation may be neglected, and the equation of continuity be- 
comes 

2,.(i:i + ii + i!!i)+4^ = o; 

^dx dy dz y dt 

and replacing du, dv and d w, by their values from Equations (409), 
and dividing by 2), we find 

from which, and Eq. (408), the equation connecting the extraneous 
forces with the co-ordinates xyz, and that expressive of Mariottc's 
law, the function 9 may be found, then the value of D, and finally 
that of p. 

The excursions being small, if we impose the additional condi- 
tion that the molecules of the fluid arc not acted upon by extra- 



278 ELEMENTS OF ANALYTICAL MECHANICS. 

neous forces, in which case the motions can only arise from some 
arbitrary initial disturbance ; then, Equation (408), 

and by Mariotte's law, 

p = P,D = a^.D (411) 



in which 



whence, by division, 



02 ^F= A^^'. ^4^2) 



dt dfi 



(413) 



which substituted above, gives 

■ ^ ^ «2 (^ _|. ^ + ^) . . . . (414) 
dt'' \dx^ ^ dy^ ^ dz^y, ^ ^ 

From this Equation the function cp is to be determined, then the 
value of D, from Equation (410), and that of p, from either of the 
Equations (411) or (413). 

§ 256. — If the fluid be confined in a narrow tube, so that the 
motion can only take place in the direction of its axis, the co- 
ordinate axis X may be assumed to coincide with this line; in which 
case V and w will each be zero, and, Equation (409), 

dy^ ~ ' dz^ - ' 
whence Equation (414) becomes 

S=-S '■ (-) 

To integrate this, add to both members 

d^cp 
ax ' a t 



MECHANICS OF FLUIDS. 2T9 

and we shall have 

1 ,/c?(p , d(p\ a , /d(p dcp\ 

and making 

dt ^ dx 
we have 

dV _ dV ^ 

d t ~ dx ' 

and V being a function of x and t^ we have, by differ entiating^ 

dV =z — -— ' dt -\ — :; dx ; 

at dx 

dV 

or by substitutincr for -r — its value above, 
^ "^ dt ' 

dV dV 

and by integration, 

in which F' denotes any arbitrary function. 
In like manner, by subtracting 

d^(p 
a 



d t ' dx 

from both members of Equation (415), we find 
d(!> d(p -, , . 

in which /' denotes any arbitrary function. 
Whence, by addition, 

^ = ir(xi-at) + y (x - at\ 



280 ELEMENTS OF ANALYTICAL MECHANICS. 
and by subtraction, 



^ = l..r(. + at)-l-f (.-at). 



dx 2a 



2a* 



But 



whence, 



d(p d(p 

d(p = —7— * dt -i ; dx 

dt dx 



dcp = -^-F'{x-{-at)d{x + at) - --'f{x - at)d{x -at); 



2a 
and by integration, 



2a 



cp = F{x + at) ^-/{x ~ at) 



(4ie;) 



in which F and /, denote any arbitrary functions whatever, and are 
determined from the initial conditions of the question. 

This last formula is used in discussing the subject of sound, and 
the more general equations which go before are employed in devel- 
oping the principles of light and heat as well as those of the tidal 
waves of the ocean and of the atmosphere. 



EQUILIBKIUM OF FLUmS. 



§ 257. — If the fluid be at rest, then will 



(Z2.r 



dp- "^ ' ~dl^"^^' 1^-^' 



and Equations (399) become 

dp 
d X 

dp 
dy 

dp 
dz 



= JD.X; 



D.Y: 



D.Z. 



(417) 



§258. — Multiplying the first by dx, the second by c?y, the third 
by dzy and adding we find, 

dp = D{Xdx+ Ydy + Zdz); ' . . . (418) 



MECHANICS OF FLUIDS. 281 

and by integration, 

p = fD.{Xdx + Tdy -{- Zdz); . . . .(419) 

whence, in order that the value of p may be possible for any 
point of the fluid mass, the product of the density by the function 
Xdx 4- Ydy -{- Zdz, must be an exact differential of a function of 
the three independent variables rr, y, z. Reciprocally, when this condi- 
tion is fulfilled, not only will the pressure at any point become known 
by substituting its co-ordinates, but the Equations, (417), will be sat- 
isfied, and the fluid will be in equilibrio. 

§ 259. — Conceiving those points of the fluid which experience equal 
pressures to be connected by, indeed to form a surface, then in 
passing from one point to another of this surface, we shall have 
dp =z 0, and 

Xdx + Ydy + Zdz = 0, • . . . . (420) 

which is obviously the differential equation of the surface. 

Dividing this by H d s, in which i^, denotes the resultant of the 
forces which act upon any particle, and ds, the element of any 
curve upon the surface passing through the particle, we have 

xix_rdj^z^d^_ ^ ^ 

R ds ^ R ds ^ E ds ~~ ' ^ ' 

whence the resultant of the forces acting upon any one of the 
elements of a surface of equal pressure, is normal to that surface. 
This is the characteristic of what is called a level surface, which 
may be defined to be any surface which cuts at right angles the 
direction of the resultant of the forces which act upon its particles. 

§260. — If Equation (420) be integrated, we have 

f{Xdx + Ydy + Zdz) =,C, (422) 

in which C is the constant of integration. The magnitudes of this 
constant must result from the dimensions of the surface, or from 
the volume of the fluid it envelops. By giving it different and 



282 ELEMENTS OF ANALYTICAL MECHANICS. 

suitable values, we may start from a single particle and proceed out- 
wards to the boundary of the fluid, and if the successive values 
differ by a small quantity, we shall have a series of level concentric 
strata. 

The last value assigned to C must belong to the bounding sur- 
face, which is also a surface of equal pressure ; otherwise the co- 
ordinates of this surface could not satisfy Equation (420), and con- 
sequently, Equations (417) and (421), and the surface particles could 
not be in equilibrio, which would be contrary to the hypothesis. 
Every free surface of a fluid in equilibrio is, therefore, a level sur 
face. 

§261.— Putting Equation (418) under the form 

d.p 



D 



= Xdx + Tdy + Zdz, (423) 



we see that whenever the second member is an exact differential, 
2? must be a function of i>, since the first member must also be an 
exact diflferential. Making, therefore, 

P = F{D), (424) 

in which F denotes any function whatever, the above equation be- 
comes 

'^-^^^ Xdx+ Ydy + Zdz', . . . (425) 

but for a level surface or stratum, the second member reduces to 
zero ; whence, 

dF{D) = 0', 

and by integration, 

F{D) = 0; 

whence, not only will each level stratum be subjected to an equal 
pressure over its entire surface, but it will also have the same 
density throughout. 

§262. — If the fluid be homogeneous and of the same temperature 
throughout, then will D be constant, and the condition of equilibrium 



MECHANICS OF FLUIDS. 283 

simply requires that the function Xdx + Ydy + Zdz^ Equation 
(419), shall be an exact differential of the three independent 
variables x, y, 2, and Avhen this is not the case, the equilibrium 
will be impossible, no matter what the shape of the fluid mass, 
and though it were contained in a closed vessel. 

But the function above referred to is, § 133, always an exact 
differential for the forces of nature, which are either attractions or 
repulsions, whose intensities are functions of the distances from the 
centres through which they are exerted. And to insure the equi- 
librium, it will only be necessary to give the exterior surface such 
shape as to cut perpendicularly the resultants of the forces which act 
upon the surface particles. This is illustrated in the simple example 
of a tumbler of water, or, on a larger scale, by ponds and lakes 
which only come to rest when their upper surfaces are normal to 
the resultant of the force of gravity and the centrifugal force arising 
from the earth's rotation on its axis. 

In the case of a heterogeneous fluid subjected to the action of a 
central force, its equilibrium requires that it be arranged in concentric 
level strata, each stratum having the same density throughout. And 
the equilibrium will be stable when the centre of gravity of the 
whole is the lowest possible, § 138, and hence the denser strata should 
be the lowest. 

When the fluid is incompressible, the density may be any function 
whatever of the co-ordinates of place. It may be continuous or dis- 
continuous. When it is given, the value of the pressure is found from 
Equation (419). 

§ 263. — In compressible fluids the density and pressure are con- 
nected by law, and the former is no longer arbitrary. 
Dividing Equation (418) by Equation (389), we have 

dp_ _ Xdx + Y dy + Zdz 
P ~ P 

Integrating, 

rXdx -f Ydy + Zdz ^ , ^ 

\ogp^ J ^^^^ + log 6- . . . (42G) 



284: ELEMENTS OF ANALYTICAL MECHANICS, 
denoting the base of the Naperian systoim by e, we have 

r Xdx+Ydy+Zdz , /'y|07\ 

p=c.e-' p ' y^^^j 

and this substituted in Equation (389), gives 

rXdx-\- Ydy-\- Zdz 

D = ^f jj^ • . • . . (428) 

These equations determine the pressure and density. 

For any surface of constant pressure, the exponent of e, in Equa- 
tion (427), must be constant, its differential must, therefore, be zero, 
and all the consequences deduced from Equation (420) will follow ; 
that is, when the fluid is at rest, it must be arranged in level strata, 
each stratum having the same density throughout, with the addition 
that the law of the varying density must be continuous by the re- 
quirements of Mariotte's law. 

If the temperature vary, then will P vary, and in order that 
Equation (427) may be an exact differential, F must be a function 
of xyz, and hence, Equations (427) and (428), when p is constant, 
D will be constant ; that is, each level stratum must be of uniform 
temperature throughout. 

It is obvious that the atmosphere can never be in equilibrio ; for 
the sun heating unequally its different portions as the earth turns 
upon its axis, the layers of equal pressure, density and temperature 
can never coincide. Hence, those perpetual currents of air known as 
the trade winds, and the periodical monsoons ; also, the sea and land 
breezes, variable winds, &c., &c. 

§ 264. — Rest is a relative term ; when applied to a particle of a 
fluid mass, it means that that particle preserves unaltered its place in 
regard to the other particles; a condition consistent with a bodily 
movement of the entire mass. 

If a liquid mass turn uniformly about an axis, the preceding 
equations will make known its permanent figure. For this purpose 
it will be sufficient to join to the forces X, F", Z, the centrifugal force. 



MECHANICS OF FLUIDS. 



285 



Take the axis z as the axis of rotation ; denote the angular velocity 
by 9, and the distance of 
the particle M from the 
axis s by r; then will 



r^ = a;2 4- 2/^; 

the centrifugal force of M 
regarded as a unit of mass, 
will be 

and its components in the -^ 

direction of x and y, respectively, 



A 




'M 



r . 9^ — = x(^\ 



and these in Equation (418), give 

djp - D.{Xdx + Ydy + Zdz + of- .xdx + 9^ y . ^^y). • (429) 

When the second member is an exact differential, the permanent form 
will be possible. 

For the free surface dp =: 0, and we have 

Xdx + Ydi/-{-Zdz-{-cp^.x.dx + (p^i/dy = 0'* .(430) 
Example 1. — Let it be required to find the figure assumed by 

the free surface of a heavy and homogeneous fluid contained in au 

open vessel and rotating about a vertical axis. 
Here, 

Xz.0; r=0; Z= -p; 

and Equation (430) becomes 

gdz = (p^{xdx + ydy). 
Integrating, 



^ = f^(^' + y^)+ C: 



(431) 



which is the equation of a paraboloid whose axis is that of rotation. 



286 



ELEMENTS OF ANALYTICAL MECHANICS. 



To find the constant (7, let the vessel be a right cylinder, with 
circular base, whose radius is a, and denote by h the height due to 
the velocity of the fluid at the circumference, then 



and 



a2 92 ^ 2gh, 



(432) 



Denote by b the height of the liquid before the rotation j its 
volume will be -r a^ . h. Conceive 
the whole body of the liquid to 
be divided into concentric cylin- 
drical layers, having for a common 
axis the axis of rotation. The base 
of any one of these layers will 
have for its area, neglecting dr"^, 
^itr .dr^ and for its volume, taking 
the origin of co-ordinates in the 
bottom of the vessel, 2'rr. c?r . 0, 
which being integrated between the 
limits r = and r =r a, will give 
the whole volume of the fluid, and 
hence. 




a?b 



i: 



zr .dr + C\ 



replacing r . dr hj its value from Equation (432), and integrating 
between the limits z — C and z = h -\- (7, which are the values 
given by Equation (432) for r = and r = a^ we find 

(7=6-1/1, 

and the equation of the upper surface becomes 

The least and greatest values for ^, are b — ^h and b -\- ^h^ 
obtained by making r = and r = a, so that the depression of the 



MECHANICS OF FLUIDS. 



287 



liquid at the axis is equal to its elevation at the surface of the 
cylindrical vessel, and is equal to half the height due to the 
velocity of the latter. 



§ 2Q^.— Example 2.— Let 
the fluid elements be attract- 
ed to the centre of the mass 
by a force varying inversely 
as the square of the distance. 
Take the origin at the cen- 
tre ; denote the distance to 
the particle m from that point 
by r, and the intensity of the 
attractive force at the unit's 
distance by k. Then will 

h a 



P 

and 



cos a = — 



X = 



lex 



Z 



-m 



cos 



cos r = 



^. 7 - _ ^ . 



which in Equation (430), give 



—-{xdx-\- y dy -j- zd z) — (^"^ {x d x + y dy) = 0, 
or 

and by integration, 

making 

2,2 _|_ y2 _ ,,.2 cos2 ^, 

in which & denotes the angle made by r, with the plane x ?/, 
A J-?!..r2cos2^ = (7, 

T 4i 



288 ELEMENTS OF ANALYTICAL MECHANICS. 

and denoting the distance from the origin to the point in which 
the free surface cuts the axis z by unity, we have, by making ^ = 90°, 



A-C7- 
1 - C, 

which substituted above, and solving with respect to cos^^, gives 

icp2.cos2^ == -^^3— ^ ...... (434) 

and making r = 1 -\- u, we have 

If the angular velocity be small, then will u be very small. 
Developing the second member with this supposition, and limiting 
the terms to the first power of w, we find 

-i(p2.cos2^ = ^(m - 3z/2). . .... (434)' 

Neglecting Sic"^, and replacing u by its value, viz.: r — 1, we 
have for a first approximation, 

r = 1 4- ^ ' cos2 &. 
2k 



From Equation (434)', we find 

©2 . COS^ I 



+ 3^2^ 



2k 
and this in the equation 

r = 1 -f «, 
gives 

r = 1 + j^- . cos2 & + Su^; 

lite 

and replacing v? by its approximate value ? above, by neg- 

lecting 3«2, we have 

- , <p2 „, , S(p*.cos*5 

for the polar equation of the meridian section. 



MECHANICS OF FLUIDS. 
Comparing this with the equation 
1 



289 



= 1+1-62 cos2 d + I . e* . COS* 3 + &C., 



^/]~— e^ cos2 ^ 
they become identical by neglecting the higher powers and making 



The free surface of the fluid approximates therefore very closely 
to an ellipsoid of revolution of which the eccentricity of its meridian 
section is equal to the square root of the quotient arising from 
dividinfT the centrifugal force at the unit's distance from the axis 
of rotation, by the force of attraction at au equal distance from the 
centre. 



PEESSTJUE OF HEAVY FLUIDS. 

§266. — When a fluid contained in any vessel is acted upon by 
its own weight, if the axis z be taken vertical 
and positive downwards, then will 

X= 0; Y=0; Z = g -, 

and Equation (418) becomes, after integrating, 

p =Dgz + C; 

and assuming the plane xy to coincide with 

the upper surface of the fluid, which must, when in equilibrio, be 

horizontal, we have, by making 2 = 0, 

P' = C; 

in which p' denotes the pressure exerted upon the unit of the free 
surface. Whence, 




P 



D.g.z 



(435) 



The first member is the pressure exerted upon a unit of surface, 
every point of which unit having a pressure equal to that sustained 
by the element whose co-ordinate is z. 

19 



290 



ELEMENTS OF ANALYTICAL MECHANICS. 



If p' =: 0, then will 

p = Bgz; (436) 

and denoting by b the area of the surface pressed, and by db, the 
elijment of this surface, whose co-ordinate is s, we have, Equation 
(397), for the pressure upon this element denoted by p^, 

p^ z= Dg.z.db, 

and the same for any other element of the surface ; whence, deno- 
ting the entire pressure by P, we shall have 

F =2p^ z^ Dff.lz.db. ..... .(437) 

But if Zj denote the co-ordinate of the centre of gravity of the 
entire surface b, then will, Equations (91), 

1z .db = bZj^ 
and 

P = Dg.b.z,. (438) 

Now h z, is tbe volume of a right cylinder or prism, whose base 
is 6, and altitude z/^ D g .b .z^ is the weight of this volume of 
the pressing fluid. Whence we conclude, that the pressure exerted 
upon any surface by a heavy Jiuid is equal to the weight of a cylin- 
drical or prismatic column of the fluid whose base is equal to the 
surface pressed^ and whose altitude is equal to the distance of the cen- 
tre of gravity of the surface below the upper surface of the fluid. 

When the surface pressed is horizontal, its centre of gravity will 
be at a distance from the upper surface equal to the depth of the 
fluid. 

This result is wholly independent of the quantity of the pressing 
fluid, and depends solely upon the density of the fluid, its height, and 
the extent of the surface pressed. 

Example 1. — Required the pressure 
against the inner surface of a cubical ves- 
sel filled with water, one of its faces being 
horizontal. Call the edge of the cube a, 
the area of each face will be a^, the dis- 
tance of the centre of gravity of each 
vertical face below the upper surface will be ^c?, and that of the 






MECHANICS OF FLUIDS. 291 

lower face a; whence, the principle of the centre of gravity 
gives, 



3. 

5a2 5 



O a" 

Again, 



S = 5 a2 ; 

and these, substituted in Equation (43S), give 

P = D .g-h.z, = D.g.ZaK 

Now D g X V = D g^ is the weight of a cubic foot of water = 62,5 
lbs., whence, 

lbs. 

P = 62,5 X Sa\ 

Make a = 7 feet, then will 

lbs. 

P = 62,5 X 3 X (7)3 = 64312,5. 

The weight of the water in the vessel is 62,5 a'^, yet the pressure 
is 62,5 X ^a^, whence we see that the outward pressure to break 
the vessel, is three times the weight of the fluid. 

Example 2. — Let the vessel be a sphere filled with mercury, and 
let its radius be R. Its centre of gravity is 
at the centre, and therefore below the upper 
surface at the distance E. The surface of the 
sphere being equal to that of four of its 
great circles, we have 

b = 4'7ri22. 




whence, 



and. Equation (438), 



b.z, = 4^i2-^; 



P = ^if .D.g.R\ 



The quantity Dg X V — I> g, \^ the weight of a cubic foot of 
mercury = 843,75 lbs., and therefore, substituting the value of 
If = 3,1416, 



ib.t. 
P = 4 X 3,1416 X 843,75. i23. 



292 



ELEMENTS OF ANALYTICAL MECHANICS. 



Now suppose the radius of the sphere to be two feet, then will 
R^ =z 8, and 

lbs. lbs. 

P = 4 X 3,1416 X 843,75 x 8 = 84822,4. 

The volume of the sphere is ^':f E^; and the weight of the con- 
tained mercury will therefore be ^if E^ c/ D =: W, Dividing the 
whole pressure by this, we find 



W 



3; 



whence the outward pressure is three times the weight of the fluid. 

Example 3. — Let the vessel be a cylinder, of which the radius 
r of the base is 2, and altitude Z, 6 feet. Then will 

b,z, =z <:rr/(r -f I) =z 3,1416 X 2 x 6 x 8; 

which, substituted in Equation (438), 

F = 301,5936 X Dff, 



and 



whence, 



W= 3,1416 X 22 X 6 X Z>^ = 75,398 x Dg-, 



W 



301,5936 X Dg 



= 4; 



75,3984.2)^ 

that is, the pressure against this particular vessel is four times the 
weight of the fluid. 

§267. — The point through which the resultant of the pressure 
upon all the elements of the surface 
passes, is called the centre of i^ressure. 
Let EIF be any plane, and MN 
the intersection of this plane produced 
with the upper surface of the fluid 
which presses against it. Denote the 
area of any elementary portion n of 
the plane EIF hj db ; and let m be 
the projection of its place upon the 
upper surface of the fluid; draw m M 
perpendicular to MN, and join n with Mhy the right line n M, the 




MECHANICS OF FLUIDS. 293 

latter will also be perpendicular to M N, and the angle nMm will 
measure the inclination of the plane EIF to the surface of the 
fluid. Denote this angle by 9, the distance mn by li\ and Mn by r' \ 
then will 

h' = r' sin 9 ; 

the pressure upon the element dh^ 

D g .r sin 9 dh\ 

its moment with reference to the line MN^ 

D g r'"^ sin cp . db', 
and for the entire surface, the moment becomes 

D g .sin (p .2 r''^ db. 

Denote by r the distance of the centre of gravity of the surface 
pressed from the line M iV, its distance below the upper surface of 
the fluid will be r . sin 9 ; and the pressure upon this surface will be 

i> ^ . r sin 9 . 5 ; 

and if I denote the distance of the centre of pressure from the 
line MjV, then will 

J)g . r sin (p . b . I = D g . sin (p ."S r'^ . db, 

from which we have, 

whence, Ecjuation (264), the centre of pressure is found at the centre 
of percussion of the surface pressed. 

§268. — The principles which have just been explained, are of 
great practical importance. It is often necessary to know the pre- 
cise amount of pressure exerted by fluids against the sides of ves- 
sels and obstacles exposed to their action, to enable us so to adjust 
the dimensions of the latter as to give them sufficient strengtli to 
resist. Reservoirs in which considerable quantities of water are col- 
lected and retained till needed for purposes of irrigation, the supply 
of cities and towns, or to drive machinery ; dykes to keep the sea 



294: 



ELEMENTS OF ANALYTICAL MECHANICS, 



and lakes from inundating low districts ; artificial embankments con- 
structed along the shores of rivers to protect the adjacent country 
in times of freshets ; boilers in which elastic vapors are pent up in 
a high state of tension to propel boats and cars, and to give motion 
to machinery, are examples. 

§ 269. — As a single instance, let it be required to find the thick- 
ness of a pipe of any material necessary to resist a given pres- 
sure. 

Let A£ C he a section of pipe perpen- 
dicular to the axis, the inner surface of 
which is subjected to a pressure of p pounds 
on each superficial unit. Denote by H the 
radius of the interior circle, and by I the 
length of the pipe parallel to the axis ; 
then will the surface pressed be measured 
by 2'n' R . I; and the whole pressure by 
2'jrR.l.2o. 

By virtue of the pressure, the pipe will stretch ; its radius will 
become B -\- d B, the path described by the pressure will be dE, 
and its c^uantity of work 

2'r^B.l.pdE. 

The interior circumference before the pressure was 2'7fi?, afterwards 
2'n-[R ■}- dB), and the path described by resistance, 2'n'dB. And 
if B denote the resistance which the material of the pipe is capable 
of opposing, to a stretching force, without losing its elasticity over 
each unit of section, t the thickness of the pipe, then, by the prin- 
ciple of the transmission of work, must 




whence, 



2ir.B.l.dB.t=:2irB.I.p.dB; 
Bp 






B 



The value of p is estimated in the case of water pressure by 
the rules just given. That in the case of steam or condensed gases, 



MECHANICS OF FLUIDS. 



295 



by rules to be given presently. The value of B is readily obtained 
from Table I, giving the results of experiments on the strength of 
materials. 




EQUILIBRrUM Al^D STABILITY OF FLOATES'G BODIES. 

§ 270. — When a body is immersed in a fluid it is not only 
acted upon by its ox^ii weight, but also by the pressure arising from 
the weight of the fluid, and the circumstances of its rest or 'motion 
will be made known by Equations i^A) and {B). 

Let ED be the body ; take the plane xy m the plane of the up- 
per surface of the fluid, 
supposed at rest, and 
the axis of 2 therefore 
vertical. Denote by 
b the entire surface 
of the body, and by 
dh, one of its elements, 
whose co-ordinates of 
position are x y z. The 
pressure upon this ele- 
ment will be 

D .g .z .db^ 

in which D is the density of the fluid, and g the force of gravity. 

This pressure is, § 248, normal to the surface, and denoting by 
a, ^ and 7, the angles which this normal makes with the axes xyz^ 
respectively, the components of the pressure in the direction of these 
axes will be 

i> • ^ . 2 . cZ 6 . cos a ; I) . g .z .db.cos^ ; D .g .z.db , cos 7. 

Similar expressions being found for the components of the pressure on 
other elements, we have, by taking their sum, 

D g .1 z.d b .COS a; D g ,11 z . d b .cos (3 -, D g . :Z z . db . cosy. 

But c? 5. cos a, db. cos ft, and db. cosy, are the projections of the 
area db on the co-ordinate planes z y, z x and x y, respectively; and 



296 ELEMENTS OF Al^ALYTICAL MECHANICS. 

II z . d b . cos a, I, z . d b . cos ^^I, z .d b . cos y, are volumes of right 
prisms -whose bases are projections of the entire surface pressed 
upon the same co-ordinate planes, and of which the altitude of each 
is the depth of the common centre of gravity of the elements of its 
base submerged to the depths of their corresponding surface elements. 
Whence we conclude, that ike component of ike pressure on any 
surface^ estimated in any direction, is equal to the pressure on so much 
of that surface as is equal to its projection on a plane at right angles 
to the given direction. 

The cylinder or prism which projects an element on one side of 
the body will also project an element situated on the opposite side ; 
these projections will, therefore, be equal in extent, but will have 
contrary signs, for the normal to the one will make an acute, and 
to the other an obtuse angle with the axis of the plane of projection. 
When these projections are made upon any vertical plane, the value 
of z will be the same in both, and hence, for each positive product, 
z .db . cos a and z . db . cos /3, there will be an equal negative product ; 
therefore, 

D g .2 z .db.GOsa = :EFGOsa = 0; Dg .I^z.db . cos (3 =z2 F cos (3 = 0. 

That is, the sum of the horizontal pressures in the directions of 
cc and y, and therefore in all horizontal directions, will be zero ; and 
the first and second of Equations (120), give 

d"^ X ^ d^y 

dt^ ' di^ ' 

or, w^iich is the same thing, there can be no horizontal motion ol 
translation from the fluid pressure. 

When the projections of opposite elements are made upon a 
horizontal plane, they will still be equal with contrary signs, the 
normal to the elements on the lower side making obtuse, while the 
normals to the elements above make acute angles with the axis z, 
but the corresponding values of z will differ, and by a length equal 
to that of the vertical filament of the body of which these elements 
form the opposite bases, and hence 

J) g. 2 z. db. cos Y = I) g.i: {z'—z)db cosy = — D g I. c d b cosy -- [^AO) 



MECHANICS OF FLUIDS. 29T 

in "which z' denotes the ordinate for the upper, and z^ that for the 
lower element in the same vertical line, and c the distance betweeu 
the elements ; and the third of Equations (120) becomes 

/ (P z\ d'^ z 

2 (Pcos/ — m ' -r^J = Mg — Dg-^c-db-fiosy — 2m. — -^ = 0. 

But Jlc.db.cosy is the volume of the immersed body which is 
obviously equal to that of the displaced fluid ; also D g .'2,c .d b . cosy 
is the weight of the displaced fluid ; and Mg that of the body. 
Denoting the volume of the body by V\ its density by i)', the 
above may be written 



Now, when 



or 



then will 



d'^z 
V'D'g - V'Dg -^m'-^ = Q. . . . (441) 



V D' g - V'Dg = 0, 
D = D\ 



d-^z 

and there can be no vertical motion of translation from the fluid 
pressure and the body's weight. 
When D' > D, then will 

and the body will sink with an accelerated motion. 
When B' < D, then will 

d'^z 

and the body will rise with an accelerated motion till 

d^z 
:s:m> — =V^D'g - VDg = 0; • • • (442) 



298 



ELEMENTS OF ANALYTICAL MECHANICS. 



in which V denotes the volume ABC, of the 
fluid displaced. At this instant we have 



V'D'c/= YDg; 



(443) 




and if the body be brought to rest, it will 
remain so. That is, the body will float at the 
surface when the weight of the fluid it dis- 
places is equal to its own weight. 

Tlie action of a heavy fluid to support a body wholly or partly 
immersed in it, is called the buoyant effort. The intensity of the 
buoyant effort is equal to the weight of the Jiuid displaced. 

Substituting the values of the horizontal and vertical components 
of the pressures in Equations (118), and reducing by the relations, 



D g .'Si c .db . cos/ . x' = D g . V . x 
D g .Sc .db . cos y .y' = D g . V . y 



(444) 



in which x and y are the co-ordinates of the centre of gravity of the 
displaced fluid referred to the centre of gravity of the body, we find 



x'.d^y' - y' .d^x' 
2 m ^^ = ; 



2 m 
2 m 







df' 




z' 


'd-'x' - x' 


'd'^z' 






df^ 




y' 


f?2 2 


'' - z' 


d'^y' 



72 



^Bg^V'X', 
= -Dg>V'y. 



(445) 



Equations (444) show that the line of direction of the buoyant 
effort passes through the centre of gravity of the displaced fluid. 
This point is called the centre of buoyancy. And from Equations 
(445), we see that as long as x and y are not zero, there will be 
an angular acceleration about the centre of gravity. At the instant 
X =: and y = 0, that is to say, when the centres of gravity of 
the body and displaced fluid are on the same vertical line, this 
acceleration will cease, and if the body were brought to rest, it 
would ha\c no tendency to rotate. 

To recapitulate, we find. 



MECHANICS OF FLUIDS. 



299 



1st. That the pressures ujpon the surface of a body immersed in 
a heavy fluid have a single resultant, called the buoyant effort of the 
fluid, and that this resultant is directed vertically upivards. 

2d. That the buoyant effort is equal in intensity to the weight of 
the fluid displaced. 

3d. That the line of direction of the buoyant effort passes through 
the centre of gravity of the displaced fluid, 

4th. That the horizontal pressures destroy one another. 

§271. — Having discussed the equilibrium, consider next the sta- 
bility of a floating body. The density of the body may be homo- 
geneous or heterogeneous. 
Let A B CD be a section 
of the body by the upper 
surface of the fluid Avheii 
the body is at rest, G 
its centre of gravity, and 
H that of the fluid dis- 
placed. Denote by V the 
volume of the displaced 
fluid, and by M the mass 
of the entire body. The 

body being in equilibrio, the line 6^ 7/ will be vertical, and denoting 
the density of the fluid by i), we shall have 




M = D.V. 



(446) 



Suppose the section ABCD either raised above or depressed 
below the surface of the fluid, and at the same time slightly careened ; 
also suppose, when the body is abandoned, that the elements have 
a slight velocity denoted by u, u\ &c. Now the question of sta- 
bility will consist in ascertaining whether the body will return to its 
former position, or will depart more and more from it. 

The free surface of the fluid is called the plane of floatation^ 
and during the motion of the body this plane will cut from it a 
variable section. 

Let A' B' C D' be one of these sections at any given instant of 



300 ELEMENTS OF ANALY^TICAL MECHANICS. 

time ; A B" C D'\ another variable section of the body by a hori- 
zontal plane through the centre of gravity of the primitive section 
ABCD^ and A C the intersection of the two. Denote by ^ the 
inclination of these two sections, and by ^ the vertical distance of 
A B" CD", from the plane of floatation, which now coincides with 
A' B' C D\ this distance being regarded as negative or positive, ac- 
cording as A B" C D" is below or above the plane of floatation. 
The variable quantities ^ and ^ will be supposed very small at the 
instant the body is abandoned. Will they continue so during the 
whole time of motion ? 

From the principles of living force and quantity of work, we have, 
Equation (121), 

Jii'.dM^.'lJiXdx + Tdy + Zdz) + C, 

The forces acting are the weights of the elements dM and the verti- 
cal pressures, the horizontal pressures destroying one another ; whence, 
X =: 0, F = 0, and 

Ju-^ dM^'l J Zdz + C =2^Zz -^ C. ' ' (447) 

The force which acts upon an element above the plane of floata- 
tion is its own weight, and the force which acts upon any element 
below that plane is the difference between its own weight and that 
of the fluid it displaces; the first will be g .dM, and the second, 
g .D .d V, in which c? F is the volume of d M\ whence, 

:lZz =fg.z.dM-fgI).z.dV. • • • (448) 

But, drawing from the centre of gravity G, of the body, the perpen- 
dicular G E, to the plane of floatation A! B' C D', and denoting G E 
by 2^, we have 

Jg.z.dMzzzgMz^. 

The mtcgra] f g D . z . d V, will be divided into two parts, viz: one 
relating to the volume of the body below A B C D, or the volume 
immersed in a state of rest, and the other that comprised between 



MECHANICS OF FLUIDS. 301 

ABCD and the plane of floatation A' B' C I>\ when the body is in 
motion. Denote hjgB V^', the value of the first, in which z' 
denotes the variable distance HF, of the centre of gravity IT. of 
the volume F, from the plane of floatation A' B' C D\ And repre- 
senting for the moment by h the value of the integral j zdV^ com- 
prehended between the planes ABCD and A' B' C D', ^i) A will 
be the second part; and Equation (447) becomes 



f 



iiP-dM z=z 2g.Mz, -2gD Vz' - 2gDh + C. • • (449) 

The line G H^ being perpendicular to the plane ABCD, the angle 
which it makes with the line G U is equal to ^, and denoting the dis- 
tance G H by a, we have 

0^ = s' ± a cos ^ ; 

the upper sign being taken when the point G is below the point 
H^ and the lower when it is above. This value reduces Equation 
(449) to 

(ii? dM=±.2gDVaQos& — IgDh+C. - • • (450) 

Let us now find the integral h. For this purpose, conceive the 
area ABCD to be divided into indefinitely small elements denoted 
by d\ and let these be projected upon the plane of floatation, 
A' B' C D' . The projecting surfiices will divide the volume com- 
prised between these two sections into an indefinite number of 
vertical elementary prisms, and these being cut by a S{^ries of hori- 
zontal planes, indefinitely near each other, will give a series of ele- 
mentary volumes, each of which will be denoted by d F, and we 
shall have 

dV — dz . dX. cos & ; 

whence, for a single elementary vertical prism, 

f zdV =: fzdz.dX. cos & =: i{zY. cos &.d\', 
IQ which (2) denotes the mean altitude of the prism, and consequently 

k =: lcos& . f{zY . d X, 
which must be extended to embrace the entire surface ABCD. 



302 ELEMENTS OF ANALYTICAL MECHANICS. 

The value of (0) is composed of two parts, viz. : one comprised 
between the parallel sections A' B' C D' and AB"CD'\ and which 
has been denoted by ^ ; the other comprised between the base d X 
and the second of these planes, and which is equal to I . sin ^, de- 
noting by I the distance of dX from the intersection A G \ whence, 

(2) - ^ + Lsin^, 

in which I will be positive or negative according as o?X happens to 
be below or above the plane AB" C D". Substituting this in the 
value of A, and recollecting that ^ and ^ are constant in the inte- 
gration, we find 

h 1= ^^^ . cos & . f d\-\-^sm& COS& fldX + I sin^ &. cos & fl^dX. 

Denote by b the area of ABCI), or the value of / dX. The 
line A C passing through the centre of gravity of A B C jD, we have 
I IdX = 0. And denoting by k^ the principal radius of gyration 
of the surface b, in reference to the axis A C, 

fpdx = bk^^, 

in which the value of k^ is dependent upon the figure and extent 
of the surface ABCD, and upon the position of the line AG. 
Whence, 

h = Ib.cos^^^ + k^^sm^&). .... (451) 
Taking 

273 ' "' 1.2 



sin ^ == ^ — ^7-^ + &c ; cos ^ = 1 — r^—^ -f- &c. 



Neglecting all the terms of the third and higher orders, substitut- 
ing the value of h, and then in Equation (450) we find, after trans- 
posing and includmg the term ±2^Z> Va, in the constant G, 

fu\dM+ gJolb^^ + {bk^^ ± Va) &•'']= G. • • -(452) 

Now the value of the constant G depends upon the initial values 
of w, ^ and ^ ; but these by hypothesis are very small ; hence (7, 
must also be very small. As long as the second term of the first 



MECHAJ^'ICS OF FLUIDS. 303 

member is positive, I u^d M must remain very small, since it is essen- 
tially positive itself, and being increased by a positive quantity, 
the sum is very small. Hence ^ and ^ must remain very small. 
But when the second term is negative, which can only be when 
hk^"^ db Fa, is negative and greater than ^ — , the value ofiu'^dM 
may increase indefinitely ; for, being diminished by a quantity that 
increases as fast as itself, the difference may be constant and very 
small. Hence, ^ and S may increase more and more after the 
body is abandoned to itself, and finally it may overturn. 

The stability of the equilibrium depends, therefore, upon the sign 
of b A-^2 ± Va ; the equilibrium is always stable when this quantity is 
positive; it is unstable when it is negative and greater than b ^'^. 
The value of b k/ = / PdX, must always be positive, since all its 
elements are positive ; the value of ± Va becomes negative when 
the centre of gravity of the body is above that of the displaced 
fluid, in which case the stability requires that 

bl'^^yVa, or, V>-^- 

When the centre of gravity of the body is below that of the dis- 
pkiced fluid, the sign of Va is positive. 

Whence we conclude that the equilibrium of a body floating at 
the surface of a heavy fluid, will be stable as long as the centre 
of gravity of the body is below that of the displaced fluid ; that 
it will also be stable about all lines A C\ with reference to which 
the principal radius of gyration of the section of the body by the 
plane of floatation squared, is greater than the volume of the dis- 
placed fluid multiplied by the distance between the centres of 
gravity of the displaced fluid and that of the body, when the latter 
is in equilibrio, divided by the area of the section of the body 
by the plane of floatation. When this condition is not fulfilled, the 
equilibrium will be unstable. A ship whose centre of gravity is 
above that of the water she displaces, may overturn about her longer, 
but not about her shorter axis. 

g272. — A line BK through the centre of gravity G of the body, 



304 



ELEMENTS OF ANALYTICAL MECHANICS. 




and which is vertical when the body is in equilibrio, is called a line 
of rest. A vertical line H' M 
through the centre of gravity 
II' of the displaced fluid, is 
called a line of support The 
point M^ in which the line of 
support cuts the line of rest, 
is called the metacentre. The 
body will be in equilibrio 
when the line of rest and of 
support coincide. The equi- 
librium will be stable if the metacentre fall above the centre of 
gravity ; unstable if below. 

§273. — "When the equilibrium is stable, and the body is disturbed 
and then abandoned to the action of its own weight and that of 
the fluid pressure, it will, in its efforts to regain its place of rest, 
oscillate about this position, and finally come to rest. 

The circumstances of those oscillations about the centre of gravity 
of the body will readily result from Equations (445). 

SPECIFIC GEAVITT. 



§274. — The specific gravity of a body, is the weight of so much 
of the body, as would be contained under a unit of volume. 

It is measured by the quotient arising from dividing the weight 
of the body by the weight of an equal volume of some other sub- 
stance, assumed as a standard ; for the ratio of the weights of equal 
volumes of two bodies being always the same, if the unit of volume 
of each be taken, and one of the bodies become the standard, its 
weight will become the unit of weight. 

The term density denotes the degree of proximity among the 
particles of a body. Thus, of two bodies, that will have the greater 
density which contains, under, an equal volume, the greater number 
of particles. The force of gravity acts, within moderate limits, 
equally upon all elements of matter. The weight of a substance 



MECHANICS OF FLUIDS. 305 

is, therefore, directly proportional to its density, and the ratio of 
the weights of equal volumes of two bodies is equal to the ratio 
of their densities. Denote the weight of the first by W, its density 
by D, its volume by F, and the force of gravity by ^, then will 

W=c/.D,V', 

and denoting the like elements of the other body by W^ , I)^ and 
Vj , we have 

Dividing the first by the second, 











W 


gBV 




D V 










w, 


~ 9i^,v, 




D,vr 


and 


maki 


»g 


the 


volumes 


equal, 
W 


D 





(453) 

Now suppose the body whose weight is W^ to be assumed as the 
standard both for specific gravity and density, then will D^ be unity, 
and 

« = -^ = ^ (454) 

in which S denotes the specific gravity of the body whose density 
is D ; and from which we see, that when specific gravities and 
densities are referred to the same substance as a standard, the 
numbers which express the one will also express the other. 

§275. — Bodies present themselves under every variety of condi- 
tion — gaseous, liquid, and solid ; and in every kind of shape and of 
all sizes. The determination of their specific gravity, in every in- 
stance, depends upon our ability to find the weight of an equal 
volume of the standard. When a solid is immersed in a fluid, it 
loses a portion of its weight equal to that of the displaced fluid. 
The volume of the body and that of the displaced fluid are equal. 
Hence the weight of the body in vacuo, divided by its loss of 
weight when immersed, will give the ratio of the weights of equal 
volumes of the body and fluid ; and if the latter be taken as the 

20 



306 ELEMENTS OF ANALYTICAL MECHANICS. 

standard, and the loss of weight be made to occupy the denomi- 
nator, this ratio becomes the measure of the specific grav.'ty of the 
body immersed. For this reason, and in view of the consideration 
that it may be obtained pure at all times and places, water is 
assumed as the general standard of specific gravities and densities 
for all bodies. Sometimes the gases and vapors are referred to 
atmospheric air, but the specific gravity of the latter being known 
as referred to water, it is very easy, as we shall presently see, to 
pass from the numbers which relate to one standard to those that 
refer to the other. 

§ 276. — But water, like all other substances, changes its density with 
its temperature, and, in consequence, is not an invariable standard. 
It is hence necessary either to employ it at a constant temperature, 
or to have the means of reducing the apparent specifi.c gravities, as 
determined by means of it at different temperatures, to what they 
would have been if the water had been at the standard temperature. 
The former is generally impracticable; the latter is easy. 

Let D denote the density of any solid, and S its specific gravity, 
as determined at a standard temperature corresponding to which the 
density of the water is D^, Then, Equation (453), 

Again, if aS" denote the specific gravity of the same body, as indi- 
cated by the water when at a temperature different from the stan- 
dard, and corresponding to which it has a density i>^^, then will 

Dividing the first of these equations by the second, we have 

whence, 

S= S'-^; ' ..... (455) 

and if the density D^ , be taken as unity, 

S=S''J),,. (456) 



MECHANICS OF FLUIDS. 



SOT 



That is to say, the specific gravity of a body as determined at the 
standard temiDerature of the ivater, is equal to its specific gravity deter- 
mined at any other temperature^ multiplied by the density of the 
water corresponding to this temperature^ the density at the standard 
temperature being regarded as unity. 

To make this rule practicable, it becomes necessary to find the 
relative densities of water at different temperatures. For this pur- 
pose, take any metal, say silver, that easily resists the chemical 
action of water, and Avhose rate of expansion for each degree of 
Fahr. thermometer is accurately known from experiment ; give it 
the form of a slender cylinder, that it may readily conform to the 
temperature of the water when immersed. Let the length of the 
cylinder at the temperature of 32° Fahr. be denoted by /, and the 
radius of its base by ml\ its volume at this temperature will be, 

rfni^l'^ X I — icm"^ P- 

Let n I he the amount of expansion in length for each degree of 
the thermometer above 32°. Then, for a temperature denoted by 
i, will the whole expansion in length be 

nl X {t - 32°), 



and the entire length of the cylin- 
der will become 

l-i-n I (t-^-2'>) = l[l+n {t-S2°)]', 

which, substituted for I in the first 
expression, will give the volume 
for the temperature t, equal to 

'jfm^P[\+n(t- 32°)]3. 

The cylinder is now weighed in 
vacuo and in the water, at differ- 
ent temperatures, varying from 32° 
upward, through any desirable range, 
say to one hundred degrees. The 
temperature at each process being 
substituted above, gives the volume 






of the displaced fluid j <te weight of the displaced fluid is known 



308 



ELEMElfTS OF ANALYTICAL MECHANICS. 



from the loss of weight of the cylinder. Dividing this weight by 
the volume, gives the weight of the unit of volume of the water at 
the temperature t. It was found by Stampfer, that the weight of 
the unit of volume is greatest when the temperature is 38°.75 Fah- 
renheit's scale. Taking the density of the water at this temperature 
as unity, and dividing the weight of the unit of volume at each of 
the other temperatures by the weight of the unit of volume at this, 
38°.75, Table II will result. 

The column under the head F, will enable us to determine how 
much the volume of any mass of water, at a temperature ^, exceeds 
that of the same mass at its maximum density. For this purpose, 
we have but to multiply the volume at the maximum density by 
the tabular number corresponding to the given temperature. 



§ 277. — Before proceeding to the practical methods of finding the 
specific gravity of bodies, and to the variations in the processes 
rendered necessary by the peculiarities of the different substances, 
it will be necessary to give some idea of the best instruments em- 
ployed for this purpose. These are the Hydrostatic Balance and 
Nicholson's Hydrometer. 

The first is similar in principle and form to the common balance. 
It is provided with numerous 
weights, extending through a 
wide range, from a small 
fraction of a grain to several 
ounces. Attached to the un- 
der surface of one of the 
basins is a small hook, from 
which may be suspended 
any body by means of a 
thin platinum wire, horse- 
hair, or any other delicate 
thread that will neither absorb 
nor yield to the chemical ac- 
tion of the fluid in which it may be desirable to immerse it. 

Nicholson's Hydrometer consists of a hollow metalic ball Aj through 




MECHANICS OF FLUIDS. 



309 




the centre of ^yhich passes a metallic wire, prolonged in both di- 
rections beyond the surface, and supporting 
at either end a basin B and B'. The 
concavities of these basins are turned in 
the same direction, and the basin B' is 
made so heavy that when the instrument 
is placed in water the stem C C shall be 
vertical, and a weight of 500 grains being 
placed in the basin B, the whole instrument 
will sink till the upper surface of distilled 
water, at the standard temperature, comes to 
a point C marked on the upper stem near 

its middle. This instrument is provided with weights similar to 
those of the Hydrostatic Balance. 

§278. — (1). If the body he solid, insoluble in water, and, will sink 
in that fiuid, attach it, by means of a hair, to the hook of the 
basin of the hydrostatic balance ; counterpoise it by placing weights 
in the opposite scale ; now immerse the body in water, and restore 
the equilibrium by placing weights in the basin above the body, 
and note the temperature of the water. Divide the weights in the 
basin to which the body is not attached by those in the basin to 
which it is, and multiply the quotient by the density corresponding 
to the temperature of the water, as given by the table ; the result 
will be the specific gravity. 

Thus denote the specific gravity by >S', the density of the water 
by D^^ , the weight in the first case by W, and that in the scale 
above the solid by iv^ then will 



(2). If the body he insoluble, hut will not sink in water, as would 
be the case with most varieties of wood, wax, and the like, attach 
to it some body, as a metal, whose weight in the air and loss of 
weight in the water are previously found. Then proceed, as in the 
case before, to find the weights which will counterpoise the com- 
pound in air and restore the equilibrium of the balance when it is 



310 ELEMENTS OF ANALYTICAL MECHANICS. 

immersed in the water. From the weight of the compound in air, 
subtract that of the denser body in air ; from the loss of weight 
of the compound in water, subtract that of the denser body ; 
divide the first difference by the second, and multiply by the density 
of the water answering to its temperature, and the result will be 
the specific gravity sought. 

£lxample» 

grs. 

A piece of wax and copper in air = 438 =z W -j- W\ 

Lost on immersion in water - - =z 95,8 = w -\- w\ 

Copper in air ------- = 388 =: W\ 

Loss of copper in water - » - = 44,2 = w\ 

Then 

TT + TF^ - TF' = 438 - 388 = 50, = W, 
^ ^ ^0^ ^ w' = 95,8 - 44,2 = 51,6 = w. 

Temperature of water 43°,25, 

D,, = 0,999952, 

W 50 

S = D^, X — = 0,999952 x -— = 0,968. 
w 51,6 

(3). J/^ the body rendihj dissolve in water, as many of the salts, 
sugar, &;c., find its apparent specific gravity in some liquid in which 
it is insoluble, and multiply this apparent specific gravity by the 
density or specific gravity of the liquid referred to water as its 
maximum density as a standard ; the product will be the true specific 
gravity. 

If it be inconvenient to provide a liquid in which the solid is 
insoluble, saturate the water with the substance, and find the appa- 
rent specific gravity with the water th-is saturated. Multiply this 
apparent specific gravity by the density of the saturated fluid, and 
the product will be the specific gravity referred to the standard. 
This is a common method of finding the specific gravity of gunpow- 
der, the water being saturated with nitre. 

(4). If the body be a liquid, select some solid that will resist its 
chemical action, as a massive piece of glass suspended from fine 



MECHANICS OF FLUIDS. 311 

platinum wire ; • weigh it in air, then in water, and finally in the 
liquid ; the differences between the first weight and each of the 
latter, will give the weights of equal volumes of water and the 
liquid. Divide the weight of the liquid by that of the water, and 
the quotient will be the specific gravity of the liquid, provided the 
temperature of water be at the standard. If the water have not 
the standard temperature, multiply this apparent specific gravity by 
the tabular density of the water corresponding to the actual tem- 
perature. 

Example. 

Loss of glass in water at 41°, 150 = w\ 
" " sulphuric acid, 277,5 = w^ 

S = ?^^ X 0,999988 =z 1,85. 
loO 

(5). If the body he a gas or vapor, provide a large glass flask- 
shaped vessel, weigh it when filled with the gas ; withdraw the gas, 
which may be done by means to be explained presently, fill with 
water, and weigh again ; finally, withdraw the water and exclude the 
air, and weigh again. This last w^eight subtracted from the first, 
will give the weight of the gas that filled the vessel, and subtracted 
from the second will give the weight of an equal volume of water; 
divide the weight of the gas by that of the water, and multiply 
by the tabular density of the water answering to the actual tem- 
perature of the latter ; the result will be the specific gravity of 
the gas. 

The atmosphere in which all these operations must be performed, 
varies at different times, even during the same day, in respect to 
temperature, the weight of its column which presses upon the earth, 
and the quantity of moisture or aqueous vapor it contains. That is 
to say, its density depends upon the state of the thermometer, barom- 
eter, and hygrometer. On all these accounts corrections must be 
made, before the specific gravity of atmospheric air, or that of any 
gas exposed to its pressure, can be accurately determined. The prin- 
ciples according to which these corrections are made, will be discussed 
when we come to treat of the properties of elastic fluids. 



ELEMENTS OF ANALYTICAL MECHANICS. 



To find the specific gravity of a solid by means^ of Nicholson's 
Hydometer, place the instrument in water, and add weights to the 
upper basin until it sinks to the mark on the upper stem ; remove 
the weights and place the solid in the upper basin, and add weights 
till the hydrometer sinks to the same point ; the difiference between 
the first weights and those added with the body, will give the 
weight of the latter in air. Take the body from the upper basin, 
leaving the weights behind, and place it in the lower basin ; add 
weights to the upper basin till the instrument sinks to the same point 
as before, the last added weights will be the weight of the water 
displaced by the body ; divide the weight in air by the weight of 
the displaced water, and multiply the quotient by the tabular density 
of the water answering to its actual temperature ; the result will be 
the specific gravity of the solid. 

To find the specific gravity of a fluid by this instrument, immerse 
it in water as before, and by weights in the upper basin sink it to 
the mark on the upper stem ; add the weights in the basin to the 
weight of the instrument, the sum will be the weight of the dis- 
placed water. Place the instrument in the fluid whose specific gravity 
is to be found, and add weights in the upper basin till it sinks to 
the mark as before; add these weights to the weight of the instru- 
ment, the sum will be the weight * of an equal volume of the fluid ; 
divide this weight by the , weight of the 
water, and multiply by the tabular density 
corresponding to the temperature of the 
water, the result will be the specific gravity. 

§ 279. — Besides the hydrometer of Nichol- 
son, which requires the use of weights, there 
is another form of this instrument which is 
employed solely in the determination of the 
specific gravities of liquids, and its indications 
are given by means of a scale of equal parts. 
It is called the Scale- Areometer. It consists, 
generally, of a glass vial-shaped vessel J, ter- 
minating at one end in a long slender neck (7, 



to receive the scale, and at the other in a 




MECHANICS OF FLUIDS. 313 

small globe B^ filled with some heavy substance, as lead or mercury 
to keep it upright when immersed in a fluid. The application and 
use of the scale depend upon this, that a body floating on the surface 
of different liquids, will sink deeper and deeper, in proportion as the 
density of the fluid approaches that of the body ; for when the body 
is at rest its weight and that of the displaced fluid must be equal. 
Denoting the volume of the instrument by F, that of the dis- 
placed fluid by F', the density of the instrument by 2>, and that 
of the fluid by D\ we must always have 

in which g denotes the force of gravity, the first member the weight 
of the instrument, and the second that of the displaced fluid. Divi- 
ding both members by D' F, and omitting the common factor g 
we have 

^ _ F 
IJ' ~ T' 

In which, if the densities be equal, the volumes must be equal; 
if the density D' of the fluid be greater than i), or that of the 
solid, the volume F of the solid must be greater than F', or that 
of the displaced fluid ; and in proportion as D' increases in respect 
to i>, will V diminish in respect to F; that is, the solid will 
rise higher and higher out of the fluid in proportion as the den- 
sity of the latter is increased, and the reverse. The neck C of 
the vessel should be of the samxC diameter throughout. To estab- 
lish the scale, the instrument is placed in distilled water at the 
standard temperature, and when at rest the place of the surface 
of the water on the neck is marked and numbered 1 ; the instru- 
ment is then placed in some heavy solution of salt, whose specific 
gravity is accurately known by means of the Hydrostatic Balance, 
and when at rest the place on the neck of the fluid surflice is af^ain 
marked and characterized by its appropriate number. The same pro- 
cess being repeated for rectified alcohol, will give another point 
towards the opposite extreme of the scale, which may be completed 
by graduation. 



814 ELEMENTS OF ANALYTICAL MECHANICS. 

To use this instrument, it will be sufficient to immerse it in a 
fluid and take the number on the scale which coincides with the 
surface. 

To ascertain the circumstances which determine the sensibility 
both of the Scale-Areometer and Nicholson's Hydrometer, let s de- 
note the specific gravity of the fluid, c the volume of the vial, I the 
length of the immersed portion of the narrow neck, r its semi-diame- 
ter, and w the total weight of the instrument. Then will 'K r^, denote 
the area of a section of the neck, and -n' r^ /, the volume of fluid dis- 
placed by the immersed part of the neck. The weight, therefore, of 
the whole fluid displaced by the vial and neck will be 

8C + S'Ji r'^l ; 

but this must be equal to the weight of the instrument, whence, 

w =: s{c -{• 'Kr'^l)^ 



from which we deduce 



5 

W 



c -f rtr'^V 
w — sc 



TC V^ S 



(457) 



Now, immersing the instrument in a second fluid whose specific gravi- 
ty is s', the neck will sink through a distance V, and from the last 
equation we have 



.2o' ' 



subtracting this equation from that above and reducing, we find 

It r^ \ s s / 

The difference I — l' is the distance between two points on the scale 
which indicates the diflference s' — s of specific gravities, and this 
we see becomes longer, and the instrument more sensible, therefore, 
in proportion as to is made greater and r less. Whence we con- 
clude that the Areometer is the more valuable in proportion as the 
vial portion is made larger and the neck smaller. 



MECHANICS OF FLUIDS. 315 

If the specific gravity of the fluid remain the same, which is the 
case with Nicholson's Hydrometer, and it becomes a question to 
know the effect of a small weight added to the instrument, denote 
this weight by w', then will Equation (457) become 

„ w + lo' — sc 



subtracting from this Equation (457), we find 

From which we see that the narrower the upper stem of Nicholson's 
instrument, the greater its sensibility. 

The knowledge of the specific gravities or densities of different 
substances, Table III, is of great importance, not only for scientific 
purposes, but also for its application to many of the useful arts. 
This knowledge enables us to solve such problems as the follow 
ing, viz. : — 

1st. The weight of any substance may be calculated, if its volume 
and specific gravity be known. 

2d. The volume of any body may be deduced from its specific 
gravity and weight. Thus we have always 

in which ff is the force of gravity, D the density, V the volume, 
and W the weight, of which the unit of measure is the weight of 
a unit of volume of water at its maximum density. 

Making D and V equal to unity, this equation becomes 

but if the density be one, the substance must be water at 3S°,75 

Fahr. The weight of a cubic foot of water at 60° is G2,5 lbs., and, 

therefore, at 38°, 75, it is 

lbs: 
62,5 »:'■ 

0;99914 = ^^""'*'; 
whence, if the volume be expressed in cubit feet, 

lbs. 

W = G2,55G X D V (458) 



316 



ELEMENTS OF ANALYTICAL MECHANICS, 



in which W is expressed in pounds ; and if the unit of volume be 
a cubic inch, 



Also, 



W = "^^^ DV = 0,036201 B V, 



V = 



w: 



62,556 . D 



W. 



lbs. 

0,036201 . D 



(459) 
(460) 

(461) 



Example 1. — Required the weight of a block of dry fir, containing 
50 cubic inches. The specific gravity or density of dry fir is 0,555, 
and F =: 50 ; substituting these values in Equation (459), 

W = 0,036201 X 0,555 X 50 = 1,00457. 

Example 2. — How many cubic inches are there in a 12-pound 
cannon-ball- 1 Here B^^ is 12 pounds, the mean specific gravity of 
cast iron is 7,251, which, in Equation (461), give 

12 



0,036201 X 7,251 



45,6. 



ATMOSPHERIC PRESSURE. 

§280. — The atmosphere encases, as it were, the whole earth. It 
has weight, else the repulsive action among its own particles would 
cause it to expand and extend itself through space. The weio-ht of 
the upper stratum of the atmosphere is in equilibrio with the re- 
pulsive action of the strata below it, and this condition determines 
the exterior limit. 

Since the atmosphere has weight, it must 
exert a pressure upon all bodies within it. 
To illustrate, fill with mercury a glass tube, 
about 32 or 33 inches long, and closed at 
one end by an iron stop-cock. Close the 
open end by pressing the finger against it, 
and invert the tube in a basin of mercury; 
remove the finger, the mercury will not 
escape, but remain apparently suspended, at 




MECHANICS OF FLUIDS. 317 

the le\'el of the ocean, nearly 30 inches above the surface of the 
mercury in the basin. 

The atmospheric air presses on the mercury with a force sufficient 
to maintain the quicksilver in the tube at a height of nearly 30 
inches ; whence, the intensity of its pressure must be equal to the weight 
of a column of mercury whose base is equal to that of the surface 
pressed and whose altitude is about 30 inches. The force thus exerted^ 
is called the atmospheric pressure. 

The absolute amount of atmospheric pressure was first discovered 
by Torricelli, and the tubes employed in such experiments are called, 
on this account, Torricellian tubes, and the vacant space above the 
mercury in the tube, is called the Torricellian vacuum. 

The pressure of the atmosphere at the level of the sea, support- 
ing as it does a column of mercury 30 inches high, if we suppose 
the bore of the tube to have a cross-section of one square inch, 
the atmospheric pressure up the tube will be exerted upon this 
extent of surface, and will support 30 cubic inches of mercury. 
Each cubic inch of mercury weighs 0,49 of a pound — say half a 
pound — from which it is apparent that the surfaces of all bodies^ at 
the level of the sea, are subjected to an atmospheric pressure of fifteen 
pounds to each square inch. 

BAEOMETEE. 

§281. — The atmosphere being a heavy and elastic fluid, is com- 
pressed by its own weight. Its density cannot be the same through- 
out, but diminishes as we approach its upper limit where it is least, 
being greatest at the surface of the earth. If a vessel filled with 
air be closed at the base of a high mountain and afterwards opened 
on its summit, the air will rush out ; and the vessel being closed 
again on the summit and opened at the base of the mountain, the 
air will rush in. 

The evaporation which takes place from large bodies of water, 
the activity of vegetable and animal life, as well as vegetable decom- 
positions, throw considerable quantities of aqueous vapor, carbonic 
acid, and other foreign ingredients temporarily into the permanent 



318 



ELEMENTS OF ANALYTICAL MECHANICS. 



portions of the atmosphere. These, together with its ever-varying 
temperature, keep the density and elastic force of the air in a 
state of almost incessant change. These changes are indicated by 
the Barometer^ an instrument employed to measure the intensity of 
atmospheric pressure, and frequently called a weather-glass^ because 
of certain agreements found to exist between its indications and the 
state of the weather. 

The barometer consists of a glass tube about thirty-four or thirty- 
five inches long, open at one end, partly filled with distilled mer- 
cury, and inverted in a small cistern also containing mercury. A 
scale of equal parts is cut upon a slip of metal, and placed against 
the tube to measure the height of the mercurial column, the zero 
being on a level with the surface of the mercury in the cistern. 
The elastic force of the air acting freely upon the mercury in the 
cistern, its pressure is transmitted to the interior of the tube, and 
sustains a column of mercury whose weight it is just sufficient to 
counterbalance. If the density and consequent elastic 
force of the air be increased, the column of mercury 
will rise till it attain a corresponding increase of 
weight; if, on the contrary, the density of the air 
diminish, the column will fall till its diminished 
weight is sufficient to restore the equilibrium. 

In the Common Barometer, the tube and its cis- 
tern are partly inclosed in a metallic case, upon 
which the scale is cut, the cistern, in this case, hav- 
ing a flexible bottom of leather, against which a 
plate a at the end of a screw b is made to press, 
in order to elevate or depress the mercury in the 
cistern to the zero of the scale. 

, De Luc's Siphon Barometer consists of a glass 
tube bent upward so as to form two unequal par- 
allel legs : the longer is hermetically sealed, and 
constitutes the Torricellian tube ; the shorter is open, 
and on the surface of the quicksilver the pressure 
of the atmosphere is exerted. The difference be- 
tween the levels in the longer and shorter legs is the barometric 



31 

30 
J2D 


■ 




H 


■ 


= 


^^^ 



bJ 



MECHANICS OF FLUIDS. 



319 



SI— 1- 

50 




height. The most convenient and practicable way of measuring this 
difference, is to adjust a movable scale between 
the two legs, so that its zero may be made to 
coincide with the level of the mercury in the 
shorter leg. 

Different contrivances have been adopted to ren- 
der the minute variations in the atmospheric pres- 
sure, and consequently in the height of the barome- 
ter, more readily perceptible by enlarging the di- 
visions on the scale, all of which devices tend to 
hinder the exact measurement of the length of the 
column. Of these we may name Morland's Diago- 
nal, and Hook's Wheel-Barometer, but especially 
Huygen's Double-Barometer. 

The essential properties of a good barometer, 
are : width of tube ; purity of the mercury ; accu- 
rate graduation of the scale ; and a good vernier. 

§282. — The barometer may be used not only to measure the 
pressure of the external air, but also to determine the density and 
elasticity of pent-up gases and vapors. When thus employed, it is 
called the harometer-gauge. In every case it will 
only be necessary to establish a free connection 
between the cistern of the barometer and the vessel 
containing the fluid whose elasticity is to be indi- 
cated ; the height of the mercury in the tube, 
expressed in inches, reduced to a standard tempera- 
ture, and multiplied by the known weight of a 
cubic inch of mercury at that temperature, will 
give the pressure in pounds on each square inch. 
In the case of the steam in the boiler of an en- 
gine, the upper end of the tube is sometimes left 
open. The cistern ^ is a steam-tight vessel, partly 
filled with mercury, a is a tube communicating 
with the boiler, and througli which the steam flows 
and presses upon the mercury ; the barometer tube 
6 c, open at top, reaches nearly to the bottom of the vessel A, 




320 



ELEMENTS OF ANALYTICAL MECHANICS. 



having attached to it a scale whose zero coincides with the level 
of the quicksilver. On the right is marked a scale of inches, and 
on the left a scale of atmospheres. 

If a very high pressure were exerted, one of several atmospheres 
for example, an apparatus thus constructed would 
require a tube of great length, in which case Ma- 
riotte's manometer is considered preferable. The tube 
being filled with air and the upper end closed, the 
surface of the mercury in both branches will stand 
at the same level as long as no steam is admitted. 
The steam being admitted through o?, presses on the 
surface of the mercury a and forces it up the branch 
be, and the scale from 6 to c marks the force of 
compression in atmospheres. The greater width of 
tube is given at a, in order that the level of the 
mercury at this point may not be materially affected 
by its ascent up the branch be, the point a being the zero of the 
scale. 




§283. — Another very important use of the barometer, is to find 
the difference of level between two places on the earth's surface, as 
the foot and top of a hill or mountain. 

Since the altitude of the barometer depends on the pressure of 
the atmosphere, and as this force depends upon the height of the 
pressing column, a shorter column will exert a less pressure than a 
longer one. The quicksilver in the barometer falls when the instru- 
ment is carried from the foot to the top of a mountain, and rises 
again when restored to its first position : if taken down the shaft 
of a mine, the barometric column rises to a still greater height. At 
the foot of the mountain the whole column of the atmosphere, from 
its utmost limits, presses with its entire weight on the mercury; 
at the top of the mountain this weight is diminished by that of 
the intervening stratum between the two stations, and a shorter 
column of mercury will be sustained by it. 

It is well known that the surface of the earth is not uniform, 
and does not, in consequence, sustain an equal atmospheric pressure 



MECHANICS OF FLUIDS. 321 

at its different points; whence the mean altitude of the barometric 
column will vary at different places. This furnishes one of the 
best and most expeditious means of getting a profile of an extended 
section of the earth's surface, and makes the barometer an instru- 
ment of great value in the hands of the traveller in search of 
geographical information. 

§ 284. — To find the relation which subsists between the altitudes 
of two barometric columns, and the difference of level of the points 
where they exist, resume Equation (427). The only extraneous force 
acting being that of gravity, we have, taking the axis z vertical, 
and counting z positive upwards, 

X= 0; F= 0; Z= -g. 

and hence, 

;p = Ce'^T (462) 

Making « = 0, and denoting the corresponding pressure by ^^, we find 

V. = ^; 

and dividing the last Equation by this one, 

V _Ci 
V, 
whence, denoting the reciprocal of the common modulus by Jf, 

, = ^^.log^ (463) 

9 P 

Denote by h^ and ^, the barometric heights at the lower and upper 
stations, respectively, then will 

II - hi. 

p ~ li' 

and reducing the barometric column h to what it would have been 
had the temperature of the mercury at the upper not differed from 
that at the lower station, by Equation (394), we have 

Pj_ ^ ^y 

p /i [1 -(- (r - T') .0,0001001]' 

in which T denotes the temperature of the mercury at *iie lower and 

T that at the upper station. 

21 



322 ELEMENTS OF AITALYTICAL MECHANICS. 

Moreover, Equation (381), 

^ = ^' (1 - 0,002551 cos 2 ■^) ; 

in which, 

/ 
ff' = 32,1808 = force of gravity at the latitude of 45°. 

P 
Substituting the value of — ^ • , of ^, and that of P, as given by 

Equation (393), in Equation (463), we find 

MDJ,, 1 + (^-32^)0,00208 FA 1 -\ 

D, 1-O,002551cos24.^ ^L/i l + (r-r}0,000100lj' 

In this it will be remembered that t denotes the temperature of 
the air; but this may not be, indeed scarcely ever is, the same at 
both stations, and thence arises a difficulty in applying the formula. 
But if we represent, for a moment, the entire factor of the second 
member, into which the factor involving t is multiplied, by X, then 
we may write 

z ^\\ J^ (t - 32°)0,00208] X 

If the temperature of the lower station be denoted by t^ , and this 
temperature be the same throughout to the upper station, then will 

0^ = [1 + {t^ - 32°) 0,00208] X 

And if the actual temperature of the upper station be denoted by t\ 
and this be supposed to extend to the lower station, then would 

z' = [1 -\- (t' - 32°) 0,00208] X 

Now if t^ be greater than t\ which is usually the case, then will the 
barometric column, or A, at the upper station, be greater than would 
result from the temperature t\ since the air being more expanded, 
a portion which is actually below would pass above the upper 
station and press upon the mercury in the cistern ; and because h 
enters the denominator of the value X, z^ would be too small. 
Again, by supposing the temperature the same as that at the uppei 
station throughout, then would the air be more condensed at the 
lower station, a portion of the air would sink below the upper 
station that before was above it, and would cease to act upon the 
mercurial column ^, which would, in consequence, become too small; 



MECHANICS OF FLUIDS. 323 

and this would make s' too great. Taking a mean between z, and 
z' as the true value, we find 

z = ^' \ ^' = [1 + 1 (^, -f ^' - 64°) . 0,00208] X. 



Replacing X by its value, 

^_MDJ^ l+i(^ + ^--64°)0,00208 FA, 1 T 

~ D, ' 1-0,002551 008 2+ ^LA l + (r-r)0,000100lj 

The factor ^ — '—■> we have seen, is constant, and it only re- 
mains to determine its value. For this purpose, measure with 
accuracy the difference of level between two stations, one at the 
base and the other on the summit of some lofty mountain, by 
means of a Theodolite, or levelling instrument — this will give the 
value of z ; observe the barometric column at both stations — this 
will give h and h^ ; take also the temperature of the mercury at 
the two stations — this will give T and T' ; and by a detached 
thermometer in the shade, at both stations, find the values of 
t^ and t'. These, and the latitude of the place, being substituted in 
the formula, every thing will be known except the co-efficient in 
question, which may, therefore, be found by the solution of a simple 
equation. In this way, it is found that 

^^^^^'" ~ 60345,51 English feet; 

which will finally give for sr, 

^^„ /'• 14-l(i; + <'-64°)0,00208 , Vh^ 1 -] 

.^60345,51. , -A 0,002551 cos-2-^^^^gLr^ + (r-nO,000100Tj 

To find the difference of level between any two stations, the lati- 
tude of the locality must be known ; it will then only be necessary 
to note the barometric columns, the temperature of the mercury, 
and that of the air at the two stations, and to substitute these 
observed elements in this formula. 

Much labor is, however, saved by the use of a table for the 
computation of these results, and we now proceed to explain how it 
may be formed and used. 



^24 ELEMENTS OF ANALYTICAL MECHANICS. 

Make 

60345,51 [1 + {t^ + f -64°)0,00104] = A, 



' 



Then will 



1 - 0,002551 cos 2 4. 

1 

I +{T - T) 0,0001 



= 0. 



AB 'log 



h 
z = AB' [log C + log h^ - log A] ; 

and taking the logarithms of both members, 

\ogz rzz log^ + log^ + log [log C 4- log A, — log A] . . (464) 

Making t^ + t' to vary from 40° to 162°, which will be sufficient 
for all practical purposes, the logarithms of the corresponding values 
of A are entered in a column, under the head A^ opposite the 
values tj + t\ as an argument. 

Causing the latitude 4. to vary from 0° to 90°, the logarithms 
of the corresponding values of B are entered in a column headed 
B^ opposite the values of 4'* 

The value of T — T' being made, in like manner, to vary 
from — 30° to + 30°, the logarithms of the corresponding values 
of C are entered under the head of (7, and opposite the values of 
T — T'. In this way a table is easily constructed. Table IV was 
computed by Samuel Howlet, Esq., from the formula of Mr. Francis 
Baily, which is very nearly the same as that just described, there 
being but a trifling difference in the co-efficients. 

Taking Equation (464) in connection with Table IV, we have this 
rule for finding the altitude of one station above another, viz. : — 

Take the logarithm of the barometric reading at the lower station^ 
to which add the number in the column headed C, opposite the ob- 
served value of T — T\ and subtract from this sum the logarithm 
of the barometric reading at the upper station ; talce the logarithm 
of this difference, to which add the numbers in the columns headed 
A and B, corresponding to the observed values of t^ -\- t' and -^ ; 
the sum will be the logarithm of the height in English feet. 



MECHANICS OF FLUIDS. 325 

Example. — At the mountain of Guanaxuato, in Mexico, M. Hum- 
boldt observed at the 

Upper Station. Lower Station. 

Detached thermometer, t' = 70°,4 ; t, = 77°,6. 
Attached » r = 70,4 ; T zz- 77,6. 

Barometric column, h = 23,66 ; h^ = 30,05. 

What was the difference of level ] 
Here 

t^ ^ t' = 148° ; T — T' - 7°,2 ; Latitude 21°. 

in. 

To log 30,05 = 1,4778445 

Add C for 7°,2 = 0,0003165 
1,4781610 

in. 

Sub. log 23,66 = L3740147 
Log of - - - 0,1041463 = — 1,0176439 
Add A for 148° - - - - = 4,8193975 
Add ^ for 21° - - - - = 0,0008689 
6885^1 3,8379103; 

whence the mountain is 6885,1 feet high. 

It will be remembered that the final Equation (464) was deduced 
on the supposition that the air is in equilibrio — that is to say, 
when there is no wind. The barometer can, therefore, only be used 
for levelling purposes in calm weather. Moreover, to insure accu- 
racy, the observations at the two stations whose difference of level 
is to be found, should be made simultaneously, else the temperature 
of the air may change during the interval between them ; but with 
a single instrument this is impracticable, and we proceed thus, viz. : 
Take the barometric column, the reading of the attached and detached 
thermometers, and time of day at one of the stations, say the 
lower; then proceed to the upper station, and take the same 
elements there ; and at an equal interval of time afterward, observe 
these elements at the lower station again ; reduce the mercurial 
columns at the lower station to the same temperature by Equation 
(394), take a mean of these colunms, and a mean of the tempera- 
tures of the air at this station, and use these means as a single 



ELEMENTS OF ANALYTICAL MECHANICS. 

set of observations made simultaneously with those at the higher 
station. 

Example. — The following observations were made to determine 
the height of a hill near West Point, N. Y. 

Upper Station. Lower Station. 

(1) (2) 

Detached thermometer, t' = 57° ; t, = 56° and 61°. 
Attached " T = 57,5 ; T = 56,5 and 63. 

Barometric column, k = 28,94 ; k, = 29,62 and 29,63. 

First, to reduce 29,63 inches at 63°, to what it would have 
been at 56°,5. For this purpose, Equation (394) gives 



h(l + T - T' X 0,0001) = 29,63 (1 - 6,5 x 0,0001) = 29,611. 
Then 

;i. = HM?. + ^M11 = 29:6155. 

t, + i' = 5S°,5 + 57°. - = 115°,5, 
T— T = 56°,5 - 57°,5 . = - p. 

in. 

To log 29,6155 = 1,4715191 

Add C for - 1° = 9,9999566 

1,4714757 

771. 

Sub. log of 28,94 = 1,4614985 
Log of - - - - 0,0099772 = - 3,9990087 
Add A for 115°,5 - - - = 4,8048112 
Add B for 41°,4 - - - - = 0,0001465 

636,65 2,8039664; 

whence the height of the hill is 636,65 English feet. 

MOTION OF HEAVY INCOMPRESSIBLE FLUIDS IN VESSELS. 

§ 285. — A heavy homogeneous liquid moving in a vessel, may 
be regarded as an assemblage of indefinitely thin strata arranged 
perpendicularly to the direction of the motion, and these strata may 



MECHANICS OF FLUIDS. 



327 



be regarded as so many solid bodies, provided we attribute to 
them the property of contracting and expanding in different direc- 
tions so as to maintain a constant volume in adapting themselves 
to the varying cross section of the vessel in which they are moving. 

Let ABCD he a vessel of 
which the axis is vertical, and 
whose horizontal sections vary 
only by insensible degrees ; sup- 
pose the fluid divided into an 
indefinite number of thin level 
strata w^hose volumes are equal 
to one another. We may sup- 
pose that at the end of each 
element of time any one stratum 
occupies the space filled by the 
stratum which preceded it at 
the commencement of this ele- 
ment. 

The horizontal velocities of the particles of the fluid may be 
disregarded, and the vertical velocity of any one of them will be 
the same as that of every other particle in the same stratum. 
The motion of the fluid will be known when we know that of any 
one stratum. 

§ 286. — Taking the axis of z vertical and positive upwards, we 
shall have, in Equations (400) and (401), 



0; 




X= 0; Y = 


.0; Z= -ff; u = 0; 


, therefore, 






I dp 1 dp 
D ' dx ~ ' D' dy~ 




1 dp dw 




D'dz^ "-^ dt' 



in which it will be recollected that w is the velocity of any one 
particle, and therefore of the stratum to which it belongs, in the 
direction of z. 



328 ELEMENTS OF ANALYTICAL MECHANICS. 

Multiplying the last equation by Ddz, and integrating, we hav^e 

//J on 
— .dz^C- . . (465) 

Take the following notation, viz. : — 

s = the variable area of the stratum whose velocity is w. 

s^ = the constant area of any determinate horizontal section of the 

vessel, as CD. 
S = the area of the section of the vessel by the upper surface of 

the liquid ; this may be constant or variable, according as the 

upper surface is stationary or movable. 
w^ = velocity of the stratum passing the section s^ at CD, at the 

time t. 
The fluid being incompressible, the same volume must pass 
every horizontal section in the same interval of time ; and hence 



or 



and 



but 






dw s, dw, ds dz 1 

, i i __ ID e • . • • 

dt ~ s ' dt ' ' dz dt 8^' 



dz WjSj 

dt~~~s 

Substituting this in the last term, and multiplying by dz, we 
have 

dw , dw. dz , ^ ^ ds 

and integrating, regarding z, and therefore s, as variable, 
fdw dw^ rdz w;^ 5,2 

which, in Equation (465), gives 



MECHANICS OF FLUIDS. 329 

To find the value of (7, let p z= P^ ^ when z = z^^ which corre- 
sponds to the section CD of the liquid ; then will 

which, subtracted from the equation above, gives 

Also, if P' denote the pressure at the upper surface corresponding 
to which z =z z\ we have 

^_P,._^,(._,).+ ^.,.^./;^-^.V-|![i-|!],4e9) 

Now z' — z^ =z h =z height of the fluid surface above the section 
CD \ whence, by substitution and transposition, 

The quantity of fluid flowing through every section in the same 
time being equal, we also have 

- Sdh = s,.w^.dt. (471) 

By means of this equation, t may be eliminated from Equation 
(470) ; then knowing the quantity of the liquid, the size and figure 

y^z' d z P^dz 
— = / — , 
z, S 'J S 

in which 5 is a function of z. 



§287. — The value of -— ^ being found from Equation (470), and 

substituted in Equation (468), this latter equation wiil give the value 
of the pressure p at any point of the fluid mass as soon as w^ be- 
comes known. 

Two cases may arise. Either the vessel may be kept constantly 
full while the liquid is flowing out at the bottom, or it may be 
suffered to empty itself. 

§288. — To discuss the case in which the vessel is always full, or 
the fluid retains the same level by being supplied at the top as fust 



830 ELEMENTS OF ANALYTICAL MECHANICS. 

as it flows out at the bottom, the quantity A must be constant, and 
Equation (471) will not be used. 
And making, in Equation (470), 



B 



o T 



« 2 

and solving with respect to d i, we have 

- = ^'^7- ••••••• (-^)- 

Now, three cases may occur. 

1st. S may be less than s^ , and C will be positive. 

2d. S may be equal to s^ , in which case will be zero. 

3d. S may be greater than s^ , when C will be negative, and this 

is usually the case in practice. 
In the first case, when C is positive, we have, by integrating Equa- 
tion (472), and supposing ^ = 0, when w^ = 0, 



whence, 



i = -7= • tan ^10^ k/— ; (473) 



..=^.ta„^.. (474) 



from which we see that the velocity of egress increases rapidly with 
the time ; it becomes infinite when 



~~A~ * ^ - 2"' 
or 

t=-^^ (475) 

When (7=0, then will the integration of Equation (472) give 

« = ^-»., • • (476) 



MECHANICS OF FLUIDS. 831 

or replacing A and B by their values, and finding the value of w^ , 

P' - P, 



('+^^^) 









whence, the velocity varies directly as the time, as it should, since 
the whole fluid mass would fall lilvc a solid body under the action 
of its own weight. 

When C is negative, the integration gives 

whence, 



6 •" -1 /B 

W. z= 






(478) 



in which e is the base of the Naperian system of logarithms = 2,718282. 
If the section S exceeds 5^ considerably, the exponent of e will 
soon become very great, and unity may be neglected in comparison 
with the corresponding power of e ; whence. 



(479) 



that is to say, the velocity will soon become constant. 

If the pressure at the upper surface be equal to that at the place 
of egress, which would be sensibly the case in the atmosphere, 
P' - P, ^ 0, and 



'F 


h(.^-^^) 


c -\ 


• '^i 



2^A_ 

1 - ^' 



(480) 



and if the opening below become a mere orifice, the fraction 

and 

w^ = yT^; (481) 



832 ELEMENTS OF ANALYTICAL MECHANICS. 

that is to say, the velocity with which a heavy liquid will issue 
from a small orifice in the bottom of a vessel, when subjected to 
the pressure of the superincumbent mass, is equal to that acquired 
by a heavy body in falling through a height equal to the depth of 
the orifice below the upper surface of the liquid. The velocities 
given by Equations (479), (480), (481), are independent of the 
figure of the vessel. 

If the velocity w^ be multiplied by the area s, of the orifice, the 
product will be the volume of fluid discharged in a unit of time. 
This is called the expense. The expense multiplied by the time of 
flow will give the whole volume discharged. 

§289. — The velocity w^ being constant in the case referred to in 
Equation (479), we shall have 

dt ' 

and Equation (468) becomes 

^ = p,-i)<,(.-.j-i>.y.(^-i), 

or, substituting the value of w^ , given by Equation (470), 

:p==P,-Da(z- z) + {Dgk + P' - F,) .'- ; . . (482) 

whence, it appears, that when the flow has become uniform, the pres- 
sure upon any stratum is wholly independent of the figure of the 
vessel, and depends only upon the area s of the stratum, its distance 

s ^ 
from the upper surface of the fluid, and upon the ratio -^. 

g 290. — If the vessel be not replenished, but be allowed to empty 
itself, h will be variable, as will also S except in the particular 
cases of the prism and cylinder. 

Making 

w^ = v^T^ (483) 



MECHANICS OF FLUIDS. 333 

in whitih H denotes the height due to the velocity of discharge : we 
have 



and, Equation (471) 



^t=-^^, (485) 

and by integration. 

t = C ^^ • / — ^= (486) 

To effect the integration, aS^ and IT must be found in terms of h. 
The relation between S and h will be given by the figure of the 
vessel. Then to find the relation between IT and h, eliminate w^ , 
d w^ , and d t from Equation (470), by the values above, and we have 

{— R + h) • dh + -^- — / H il - ^) dh = 0'. 

or, dividing by 

Sj^ p^ d z 
~S'Jo T' 



■(^.^■+') ^(^-p 



dh-^dH r-f ^.c^ A =0.(487) 

^ r^ dz ^ r^ dz ^ ' 

'Jo s 



and making 



-(■-» „ »(^+') 



^= T^TZTZ-'. Q = 



r^ dz ' ^ ^ r^ dz ' 

'Jos 'Jos 

Qdh -{- dll + BRdh = (488) 

/Rdk 

Multiplying by e , 

/ndh fRdh fRdh 

dh'Q^e -^dH-e -\- H . e xlidh = 0', 



334: ELEMENTS OF ANALYTICAL MECHANICS. 



or 



fRdh fRdk. 

dh' Q^e + d {He ) = 



and integrating 



whence, 



/jRdh fRdh 

dh. Q-e ^ He = (7; . . . . (489) 



—fRdh fRdh^ 

H =e ' fC - fdh-Q -e \ 



(490) 



The constant must result from the condition, that when H = 0, 
k must be A^ , the initial height of the fluid in the vessel. 

Thus H becomes known in terms of h, and its value substituted 
in Equation (486) will make known the time required for the fluid 
to reach any altitude h. The constant in Equation (486) must be 
determined, so that when ^ = 0, h = h^. 

§291. — The mode of solution here indicated is direct and general; 
but analysis, in its application to the motion of fluids, often pre- 
sents itself under forms which require us, in particular cases, to 
adapt the mode of solution to the peculiarities which belong to them. 
Take, for example, the case of a right cylinder or prism. Here S 
will be constant, and equal to s. 

dz h 



r<' dz _ 

Jo .<? 



S' 



Moreover, let us vsuppose P' — -^y = 0, which would be sensibly 
true were the fluid to flow into the atmosphere that rests upon its 

upper surface. Also, for the sake of abbreviation, make — = ^, 

then will 



E = 



h h ' 

and Equation (488) becomes 

B .h.dh ^ h .d H -^ {\ -k"") .H .dh:=zO. . (491) 



mecha:sics of fluids. 335 



Multiplying by h , ^ve have 

F . h'^^dh + h'^^dH + (1 - F) hT^'^dh . ^ = 0, 



Z,2 2 — ifc2 1 — fc2 



2 - X-2 
and by intem-ation, 



2.2 2 - fc2 1 _ fc2 

.h + H' h ^ C. 



2 - A-2 
Now, when A = h^ . then will If — ; whence, 



F 2 _ fc2 

K = a 



2 - k' 
and / 



Z:2 2 - A:2 1 _ k2 p 2 - fc^ 



2 - F 2 - ^-2 ' 

whence, 



2 _ fc2 ,2-h'^ 



H = 



2-F i_fc^ 



multiplying both numerator and denominator by A, 



H 



B - 2 
which substituted in Equation (486), gives 



V-Ct) ^----^m 



t = c 



/F - 2 .. dk ,^^^^ 

in which the only variable is h. 



§292. — The particular case in which P = 2, gives to this value 
for i the form of indctermination. When this occurs, we must have 
recourse to the form assumed by Equation (491), which, under this 
supposition, becomes 

^hdh -h hdH - Hdh 0; 



336 ELEMENTS OF ANALYTICAL MECHANICS. 



multiplying by h , 

2A"'c^A + hr\dH -H.h'^dh = 0, 

^•X + ^T-^' 

2\ogh + -j^ = C', 

and because H =l when h =i h^ . 

2 log A, = (7; 
whence, 

H z=.2h-\og \ 



and this, in Equation (486), gives 

^ dh 



-^'"v^/ 



\/3A.iogA 



Making — ^ = — -; this becomes 
h x^ 



t = C 






The value of C is determined by making x = I when ^ = 0. 

§293. — If the orifice be very small in comj^arison with a cross 
section of the prismatic or cylindrical vessel, then will H = h^ and 
Equation (486) gives 

t=: C ^.-V/^- 



Making i = when h = h^ , we have 

i= -^ • (V"^. - vA), (494) 

and for the time required for the vessel to empty itself, k = 0, and 



MECHANICS OF FLUIDS. ■ 337 

Now, with the same relation of the orifice to the cross section 
of the cylindrical vessel, we have, Equation (481), 



and for the volume of fluid discharged in the time t, when the 
vessel is kept full, 

and if this be equal to the contents of the vessel, 



whence, 

That is, Equation (495), the time required for a prismatic or cylin- 
drical vessel to discharge itself through a small orifice at the 
bottom is double that required to discharge an equal volume, if 
the vessel were kept full. 

§ 294. — The orifice being still small, we obtain, from Equa- 
tion (485), 

dh 9 J 

whence it appears that, for a cylindrical or prismatic vessel, the 
motion of the upper surface of the fluid is uniformly retarded. It 
will be easy to cause S so to vary, in other words, to give the 
vessel such figure as to cause the motion of the upper surface to 
follow any law. If, for example, it were required to give such figure 
as to cause the motion of the upper surface to be uniform, then 
would the first member of the above equation be constant ; and, 
denoting the rate of motion by a, we should have 

whence, 

_ s;^.2gh 

but supposing the horizontal sections circular. 



338 ELEMENTS OF ANALYTICAL MECHANICS, 

and, therefore, 



VSv^= 



whence the radii of the sections must vary as the fourth root of 
their distances from the bottom. These considerations apply to the 
construction of Clepsydras or Water Clocks. 

MOTION OF ELASTIC FLUIDS IN VESSELS. 



§295. — As in the case of incompressible, so also in that of 
elastic fluids, it is assumed that in their movement through vessels, 
they arrange themselves into parallel strata at right angles to the 
direction of the motion. The quantity of matter in each stratum 
is supposed to remain the same, while its density, which is always 
uniform throughout, may vary from one position of the stratum to 
another ; hence, the volume of each stratum may vary. 

All lateral velocity among the particles will be supposed zero ; 
and as the weight of the elements of elastic fluids is insignificant 
in comparison to their elasticity, the former will be disregarded. 
The motion will, therefore, be due only to the elastic force arising 
from some force of compression ; and as the fluid will be supposed 
to communicate freely with the air, or with a vessel partly filled 
with some other elastic fluid, this force within may be greater or less 
than it is on the exterior of the vessel. 

g296. — Assuming the axis of the vessel horizontal, take that 
line as the axis of x. 

Then, by the supposi- . 

tion above, will 



X=: 

Y =0 
Z =0 

V = 
.w = 



^ 



JS' 



MECHANICS OF FLUIDS. 339 

and Equations (400) give 

D dx \dt/ dx ^ ' 

Moreover, if we suppose the motion to have been established 
and become permanent, the velocity of a stratum as it passes any 
particular cross section of the vessel will always be constant, and 
the quantity of fluid which flows through every cross section will 
be the same. Hence the partial diflferential of u in regard to the 
time, that is, supposing x^ y, 2, to be constant, must be zero, and 
the above equation reduces to 

dp = ~ D ,u. du. 

From Mariotte's law. Equation (389), 



and by division, 






dp 1 , 


and by integration, 






\ogp = C-- —'u' 



..... (497) 

To determine the constant, let p^ be the pressure at the opening 
CDj that is, the pressure of the atmosphere, and denote by u^ the 
velocity of the fluid at this point, then will 



lOgi?; = C 



2P 



and by subtraction. 



log|^ = 2^-(«<^-«^). (498) 



Denote by s the area of any section of the vessel A' B\ at which 
the pressure is p and velocity v, by D the density of the fluid at 
this section, and by D^ that at the section CD equal to s^ . Then, 
since the quantities of fluid flowing through these sections in a unit 
of time must be equal, we have 

D . s . u = D, . s, .u.i 



340 ELEMENTS OF ANALYTICAL MECHANICS. 
but, §244, 



•whence. 



or 






p , S .U = p^SjUj 



U = J 

p.s 



which, in Equation (498), gives 



(499) 
Pj '^i^ I. ^p .s^ ^ ^ ' 

If p' denote the pressure exerted by the piston A B, and S de- 



note its area, we have 



■<^^S['-(j=4)"]----« 



whence. 



2P.log ^ 

\p'sy 



(501) 



This is the velocity with which the fluid will issue into the 
atmosphere or other fluid whose pressure on the unit of surface is p^ . 

§ 297. — The volume discharged in a unit of time is 



'2 P ' log 






while under the pressure p^ ; and under a pressure equal to that 
on the unit of surface of the piston, 
or top of a gasometer, and which would 
be indicated by a gauge, since the vol- 
umes are inversely as the pressures, 



U,S,=^,'S, 




P.log^ 



p, 



1 - r^Y 



(502) 




MECHANICS OF FLUIDS. 341 

§298. — Dividing Equation (499) by Equation (500), we have 



P 1 /Pi^.Xi 

.... (503) 






P 1 /Pj *A2 



loa £1- 1 _ (Pl^\ 

^P. \p's) 

which will give the pressure p at any section of the vessel. 



§299. — If the opening CD is very small in reference to A B, the 
velocity u^ will become, Equation (501), 



u,=j2P.\og^; (504) 

and the volume of fluid discharged in a unit of time and of a den- 
sity equal to that pressing upon the gauge, 



|^..,.,/3P.log|^; (505) 

and Equation (503) becomes 



log^ 



C-fi)- 



§ 300. — A stream flowing through an orifice is called a vein. In 
estimating the quantity of fluid discharged, it is supposed that there 
are neither within nor without the vessel any causes to obstruct the 
free and continuous flow ; that the fluid has no viscosity, and does 
not adhere to the sides of the vessel and orifice ; that the particles 
of the fluid reach the upper surface with a common velocity, and also 
leave the orifice with equal and parallel velocities. None of these 
conditions are fulfilled in practice, and the theoretical discharge must, 
therefore, difler from the actual. Experience teaches that the former 
always exceeds the latter. If we take water, for example, which is 
far the most important of the liquids in a practical point of view, 
we shall find it to a certain degree viscous, and always exhibiting a 
tendency to adhere to un unctuous surfaces with which it may be 
brought in contact. When water flows through an opening, the 



842 ELEMENTS OF ANALYTICAL MECHANICS. 

adhesion of its particles to the surface will check their motion, and 
the viscosity of the fluid will transmit this effect towards the interior 
of the vein; the velocity wijl, therefore, be greatest a; the axis of 
the latter, and least on and near its surface ; the inner particles thus 
flowing away from those without, the vein will increase in length and 
diminish in thickness, till, at a certain distance from the orifice, the 
velocity becomes the same throughout the same cross-section, which 
usually takes place at a short distance from the aperture. This 
effect will be increased by the crowding of the particles, arising from 
the convergence of the paths along which they approach the aper- 
ture, every particle, which enters near the edge, tending to pass 
obliquely across to the opposite side. This diminution of the fluid 
vein is called the veinal contraction. The quantity of fluid discharged 
must depend upon the degree of veinal contraction, and the velocity 
of the particles at the section of greatest diminution ; and any cause 
that will diminish the viscosity and cohesion, and draw the particles 
in the direction of the axis of the vein as they enter the aperture, 
will increase the discharge. 

Experience shows that the greatest contraction takes place at a 
distance from the vessel varying from a half to once the greatest 
dimension of the aperture, and that the amount of contraction de- 
pends somewhat upon the shape of the vessel about the orifice 
and the head of fluid. It is further found by experiment, that if a 
tube of the same shape and size as the vein, from the side of the 
vessel to the place of greatest contraction, be inserted into the 
aperture, the actual discharge of fluid may be accurately computed 
by Equation (478), provided the smaller base of the tube be sub- 
stituted for the area of the aperture ; and that, generally, without 
the use of the tube, the actual may be deduced from the theoretical 
discharge, as given by that equation, by simply multiplying the 
Jiheoretical discharge into a co-efflcient \viio^.e numerical value depends 
upon the size of the aperture and head of the fluid. Moreover, 
all other circumstances being the same, it is ascertained that this 
co-efficient remains constant, whether the aperture be circular, square, 
or oblong, which embrace all cases of practice, provided that in 
comparing rectangular with circular orifices, we compare the smallest 



MECHANICS OF FLUIDS. 



343 




dimension of the former with the diameter of the latter. The value 

of this co-efficient depends, therefore, when other circumstances are 

the same, upon the smallest dimension of the rectangular orifice, 

and upon the diameter of the circle, in the case of circular orifices. 

But should other circumstances, such as the head of fluid, and the 

place of the orifice, in respect to the sides 

and bottom of the vessel, vary, then will 

the co-efficient also vary. When the flow 

takes place through thin plates, or through 

orifices whose lips are bevelled externally, 

the co-efficient corresponding to given heads 

and orifices, may be found in Table V, 

provided the orifices be remote from the 

lateral faces of the vessel. This table is 

deduced from the experiments of Captain 

Lesbros, of the French engineers, and agrees 

with the previous experiments of Bossut, Michelotti, and others. 

As the orifice approaches one of the 
lateral faces of the reservoir, the contrac- 
tion on that side becomes less and less, 
and will ultimately become nothing, and the 
CO- efficient will be greater than those of the 
table. If the orifice be near two of these 
faces, the contraction becomes nothing on 
two sides, and the co-efficient will be still 
greater. 

Under these circumstances, we have the 
following rules : — Denote by C the tabular, 
and by C the true co-efficient corresponding 
to a given aperture and head ; then, if the 
contraction be nothing on one side, will 

C = 1,03 (7; 
if nothing on two sides, 

C = 1,0G C; 
if nothing on three sides, 

C = 1,12 (7; 





3M ELEMENTS OF ANALYTICAL MECHANICS. 

and it must be borne in mind, that these results and those of the 
table are applicable only when the fluid issues through holes in 
thin plates, or through apertures so bevelled externally that the 
particles may not be drawn aside by molecular action along their 
tubular contour. 

§301. — \Yhen the discharge is through tkiclc plates without bevel, 
or through cylindrical tubes whose lengths are from two to three 
times the smaller dimension of the orifice, the expense is increased, 
the mean coefficient, in such cases, augmenting, according to experi- 
ment, to about 0,815 for orifices of which the smaller dimension 
varies from 0,33 to 0,66 of a foot, under heads which give a coeffi- 
cient 0,619 in the case of thin plates. The cause of this increase is 
obvious. It is within the observation of every one, that water will 
wet most surfaces not highly polished or covered with an unctuouo 
coating — in other words, that there exists between the particles of 
the fluid and those of solids an affinity which will cause the former 
to spread themselves over the latter and adhere with considerable 
pertinacity. This affinity becoming eflfective between the inner sur- 
face of the tube and those particles of the fluid which enter the 
orifice near its edge, the latter will not only be drawn aside from 
their converging directions, but will take with them, by the force of 
viscosity, the other particles, with which they are in sensible contact. 
The fluid filaments leading through the tube will, therefore, be more 
nearly parallel than in the case of orifices through thin plates, the con- 
traction of the vein will be less, and the discharge consequently 
greater. 



PAET III 



APPLICATION" OF THE PRECEDING PRINCIPLES TO 
SIMPLE MACHINES, PUMPS, ETC. 

§ 302. — Any device by which the action of a force may be received 
at one place and transmitted to another is called a Machine. 

There are usually seven elementary machines discussed in Me- 
chanics ; viz., the Cord^ Lever ^ Inclined Plane^ Pulley^ Screw, Wheel and 
Axle, and Wedge. The Cord, Lever, and Inclined Plane are called 
Simple Machines ; the others, being combinations of these, are called 
Compound Machines. 

-§303. — In Machines, as in all other .bodies, every action is ac- 
companied by an equal and contrary reaction. A force which acts 
upon a Machine to impress or preserve motion is called a Power. 
A force which reacts to prevent or destroy motion; is called a 
Resistance. The Agent which is the source of power, is, § 38, called 
a Motor. 

§ 304. — Resuming Equation (30), and supposing the displacement, 

which in that equation was wholly arbitrary, to conform in every 

respect to that caused by the powers and resistances, we bhal] have 

d s = d s, s being the path described by the elementary mass m j 

and hence, 

d'^s 
2 FSp — 2 wi . — - .ds z= 0; 
d t 



but 



"whence, 



d^s . ds d^s , , , o 

— — - a s =z -— • — — - = V dv =: ^ dv^ 
dt^ dt dt ^ 



UPSp — l^lm.dv^ = 0. (50G) 



346 ELEMENTS OF ANALYTICAL MECHANICS. 

Denoting by Q, Q\ &;c. the resistances, by P, P', &;c. the pow- 
ers, S q, &LQ. and dp, &c. the projections of their respective virtual 
velocities ; the first term, which embraces all the forces except 
inertia in action on the machine, may be replaced by 2FSp — 2 QSq^ 
and we have 

^Pdp — :EQSq = ^^m.dv\ .... (507) 
Integrating, 

f^FSp - f^QSq = i^mv^ + C', 

and denoting by v^ the initial velocity, and taking the integral so 
as to vanish when ^ = 0, 

flPdp — fl Q6q = i2mv2 - lUmv^^. • • • (508) 

The products F6p and QSq are the elementary quantities of 
work performed by a power and a resistance respectively, in 
the element of time dt] the product ^mdv^ is the elementary 
quantity, of work performed by the inertia, or one half the incre 
ment of living force of the mass m in this time. And Equation 
(508) shows that in any machine, in motion, the increment of the 
half sum of the living forces of all its parts is always equal to 
the excess of the work of the powers or motors over that of the 
resistances. 

§305. — If the machine start from 'rest. Equation (508) becomes 

fi:FSp—f2QSq = l:Smv%' • • • (509) 

and as the second member is essentially positive, the work of the 
motors must exceed that of the resistances embraced in the term 
1 1 Qdq ', in other words, the inertia will oppose the motor and 
act as a resistance. When the motion becoines uniform, the second 
member will be constant ; from that instant inertia will cease to 
act, and the subsequent work of the motor will be equal to that 
of the resistances as long as this motion continues. If the motion 
be now retarded, the second member will decrease, the inertia will 
act with the power, and this will continue till the machine cornea 



APPLICATIOIITS. 347 

to rest, and the excess of work of the Resistance durmg retardation 
will be exactly equal to that of the Power during acceleration. 
Generally, then, when a machine is at rest or is moving uniformly, 
inertia does not act ; when the motion is variable, it does, and 
opposes or aids the motor according as the motion is accelerated 
or retarded. 

§306. — The essential parts of every machine are those which 
receive directly the action of the motor, those which act directly 
upon the body to be moved or transformed, and those which, serve 
to transmit the action. The arrangement of the latter is often a 
source of resistance, arising from Friction^ Adhesion^ Stiffness of 
Cordage^ &c., whose work enters largely into the general term 

FRICTION. 

§307. — When two bodies are pressed together, experience shows 
that a certain effort is always required to cause one to roll or slide 
along the other. This arises almost entirely from the inequalities in 
the surfaces of contact interlocking with each other, thus rendering 
it necessary, when motion takes place, either to break them off, com- 
press them, or force the bodies to separate far enough to allow them 
to pass each other. This cause of resistance to motion is called fric- 
tion^ of which we distinguish two kinds, according as it accompanies 
a sliding or rolling motion. The first is denominated sliding, and 
the second rolling friction. They are governed by the same laws ; 
the former is much greater in amount than the latter under given 
circumstances, and being of more importance in machines, will prin- 
cipally occupy our attention. 

The intensity of friction, in any given case, is measured by the 
force exerted in the direction of the surface of contact, which will 
place the bodies in a condition to resist, during a change of state, 
in respect to motion or rest, only by their inertia. 

§308. — The friction between two bodies maybe measured directly 
by means of the spring balance. For this purpose, let the surface 




34:8 ELEMENTS OF ANALYTICAL MECHANICS. 

C D of one of the bodies M be made perfectly level, so that the 

other body M\ when laid 

upon it, may press with 

its entire weight. To some 

point, as U, of the body 

M', attach a cord with a 

spring balance in the 

manner indicated in the figure, and apply to the latter a force F of 

such intensity as to produce in the body M^ a uniform motion. The 

motion being uniform, the accelerating and retarding forces must be 

equal and contrary; that is to say, the friction must be equal and 

contrary to the force F, of which the intensity is indicated by the 

balance. 

The experiments on friction which seem most entitled to confi- 
dence are those performed at Metz by M. Morin, under the orders 
of the French government, in the years 1831, 1832, and 1833. They 
were made by the aid of a contrivance, first suggested by M. Pon- 
celet, which is one of the most beautiful and valuable contributions 
that theory has ever made to practical mechanics. Its details are 
given in a work by M. Morin, entitled ^'' Nouvelles Experiences sur le 
FroitemenC Paris, 1833. 

The following conclusions have been drawn from these experi- 
ments, viz. : 

The friction of two surfaces which have been for a considerable 
time in contact and at rest is not only different in amount, but also 
in nature, from the friction of surfaces in continuous motion ; espe- 
cially in this, that the friction of quiescence is subjected to causes of 
variation and uncertainty from which the friction during motion is 
exempt. This variation does not appear to depend upon the extent 
of the surface of contact; for, with different pressures, the ratio of 
the friction to the pressure varied greatly, although the surfaces of 
contact were the same. 

The slightest jar or shock, producing the most imperceptible 
movement of the surfaces of contact, causes the friction of quies- 
cence to pass to that which accompanies motion. As every machine 
may be regarded as being subject to slight shocks, producing imper 



APPLICATIONS. 349 

ceptible motions in the surfaces of contact, the kind of friction to be 
employed in all questions of equilibrium, as well as of motions of 
machines, should obviously be this last mentioned, or that which 
accompanies continuous motion. 

The LAWS of friction which accompanies continuous motion are 
remarkably uniform and definite. These laws are : 

1st. Friction accompanying continuous motion of two surfaces, 
between which no unguent is interposed, bears a constant proportion 
to the force by which those surfaces are pressed together, whatever 
be the intensity of the force. 

2d. Friction is wholly independent of the extent of the surfaces in 
contact. 

3d. Where unguents are interposed, a distinction is to be made 
between the case in which the surfaces are simply unctuous and in 
intimate contact with each other, and that in which the surfaces are 
wholly separated from one another by an interposed stratum of the 
unguent. The friction in these two cases is not the same in amount 
under the same pressure, although the law of the independence of 
extent of surface obtains in each. When the pressure is increased 
sufficiently to press out the unguent so as to bring the unctuous sur- 
faces in contact, the latter of these cases passes into the first ; and 
this fact may give rise to an apparent exception to the law of the 
independence of the extent of surface, since a diminution of the sur- 
face of contact may so concentrate a given pressure as to remove the 
unguent from between the surfaces. The exception is, however, but 
apparent, and occurs at the passage from one of the cases above- 
named to the other. To this extent, the law of independence of the 
extent of surface is, therefore, to be received with restriction. 

There are, then, three conditions in respect to friction, under 
which the surfaces of bodies in contact may be considered to exist, 
viz.: 1st, that in which no unguent is present; 2d, that in which 
the surfaces are simply unctuous; 3d, that in which there is an 
interposed stratum of the unguent. Throughout each of these states 
the friction which accompanies motion is always proportional to the 
pressure, but for the same pressure in each, very difierent in 
amount. 



350 



ELEMENTS OF ANALYTICAL MECHANICS. 



4th. The friction which accompanies motion is always independ- 
ent of the velocity with which the bodies move; and this, whether 
the surfaces be without unguents or lubricated with water, oils, 
grease, glutinous liquids, .syrups, pitch, &c., &c. 

The variety of the circumstances under which these laws obtain, 
and the accuracy with which the phenomena of motion accord with 
them, may be inferred from a single example taken from the first 
set of Morin's experiments upon the friction of surfaces of oak, 
whose fibres were parallel to the direction of the motion. The sur- 
faces of contact were made to vary in extent from 1 to 84; the 
forces which pressed them together from 88 to 2205 pounds ; and 
the velocities from the slowest perceptible motion to 9,8 feet a 
second, causing them to be at one time accelerated, at another 
uniform, and at another retarded ; yet, throughout all this wide 
range of variation, in no instance did the ratio of the pressure to 
the friction differ from its mean value of 0,478 by more than gV 
of this same fraction. 

Denote the constant ratio of the normal pressure P, to the en- 
tire friction F^ by/; then will the first law of friction be expressed 
by the following equation, 

J-=/; (510) 

whence, 

F = f.P. 

This constant ratio / is called the co-efficknt of friction^ because, 
when multiplied by the total normal pressure, the product gives 
the entire friction. 

Assuming the first law of fric- 
tion, the co-efficient of friction may 
easily be obtained by means of the 
inclined plane. Let W denote the 
weight of any body placed upon 
the inclined plane A B. Eesolve 
this weight G G' into two compo- 
nents, one GM perpendicular to 
the plane, and the other G N par- 




APPLICATIONS. 351 

allel to it. Because the angles G' G M and BAC are equal, the 
first of these components will be 

GM — TT.cos^, 
and the second, 

GN — Tr.sin^, 

in -which A denotes the angle BAC. 

The first of these components determines the total pressure upon 
the plane, and the friction due to this pressure will be 

F = f. TTcos^. 

The second component urges the body to move down the plane. 
If the inclination of the plane be gradually increased till the body 
move with uniform motion, the total friction and this component 
must be equal and opposed ; hence, 

/. W . cos A = W . sin A j 
whence. 



sm A , 

f = T = tan A. 

•^ cos^ 

We, therefore, conclude, that the unit or co-efficient of friction 
between any two surfaces, is equal to the tangent of the angle 
which one of the surfaces must make with the horizon in order 
that the other may slide over it with a uniform motion, the body 
to which the moving surface belongs being acted upon by its own 
weight alone. This angle is called the anffle of friction or limiting 
angle of resistance. 

The values of the unit of friction and of the limiting angles for 
many of the various substances employed in the art of construction, 
are given in Tables VI, VII and VIII. 

The distinction between the friction of surfaces to which no un- 
guent is applied, those which are merely unctuous, and those between " 
which a uniform stratum of the unguent is interposed, appears first 
to have been remarked by M. Morin ; it has suggested to him 
what appears to be the true explanation of the difference between 
his results and those of Coulomb. He conceives, that in the ex- 



352 ELEMENTS OF ANALYTICAL MECHANICS. 

periments of this celebrated Engineer, the requisite precautions had 
not been taken to exclude unguents from the surflices of contact. 
The slightest unctuosity, such as might present itself accidentally, 
unless expressly guarded against — such, for instance, as might have 
been left by the hands of the workman who had given the last 
polish to the surfaces of contact — is sufficient materially to affect 
the co-efficient of friction. 

Thus, for instance, surfaces of oak having been rubbed with hard 
dry soap, and then thoroughly wiped, so as to show no traces 
whatever of the unguent, were found by its presence to have lost 
f*^* of their friction, the co-efficient having passed from 0,478 
to 0,164. 

This effect of the unguent upon the friction of the surfaces may 
be traced to the fact, that their motion upon one another without 
unguents was always found to be attended by a wearing of both tha 
surfaces ; small particles of a dark color continually separated from 
them, which it was found from time to time necessary to remove, 
and which manifestly influenced the friction : now, with the presence 
of an unguent the formation of these particles, and the consequent 
wear of the surfaces, completely ceased. Instead of a new surface 
of contact being continually presented by the wear, the same surface 
remained, receiving by, the motion continually a more perfect polish. 

A comparison of the results enumerated in Table VIII, leads to 
the following remarkable conclusion, easily fixing itself in the memory, 
that with the unguents^ hogs' lard and olive oil interposed in a con- 
tinuous stratum between them^ surfaces of wood on metal, wood on 
wood, metal on wood, and metal on metal, when in motion, have all 
of them very nearly the same co-efficient of friction, the value of that 
co-efficient bei7ig in all cases included between 0,07 and 0,08, and the 
limiting angle of resistance therefore between 4° and 4° 35'. 

For the unguent tallow the co-efficient is the same as the above in 
every case, except in that of metals upon metals ; this unguent seems 
less suited to metallic surfaces than the others, and gives for the 
'mean value of its co-efficient 0,10, and for its limiting angle of re- 
sistance 5° 43'. 



APPLICATIONS, 



353 



309. — Besides friction, there is another cause of resistance to the 
motion of bodies when moving over one another. The same forces 
which hold the elements of bodies together, also tend to keep the 
bodies themselves together, when brought into sensible contact. The 
effort by which two bodies are thus united, is called the force of 
Adhesion. 

Familiar illustrations of the existence of this force are furnished 
by the pertinacity with which sealing-wax, wafers, ink, chalk and 
black-lead cleave to paper, dust to articles of dress, paint to the 
surface of wood, whitewash to the walls of buildings, and the like. 

The intensity of this force, arising as it does from the affinity 
of the elements of matter for each other, must vary with the num- 
ber of attracting elements, and therefore with the extent of the sur- 
face of contact. 

This law is best verified, and the actual amount of adhesion be- 
tween different substances determined, by means 
of a delicate spring-balance. Por this purpose, 
the surfaces of solids are reduced to polished 
planes, and pressed together to exclude the air, 
and the efforts necessary to separate them noted 
by means of this instrument. The experiment 
being often repeated with the same substances, 
uaving different extent of surfaces in contact, it 
is found that the effort necessary to produce 
the separation divided by the area of the surface 
gives a constant ratio. Thus, let S denote the 
area of the surfaces of contact expressed in square 
feet, square inches, or any other superficial unit; 
A the effort required to separate them, and a 
the constant ratio in question, then will 

A 




IHIIIIIHIIl! 



or, 



a.S. 



The constant a is called the unit or co-effi,cient of adhesion^ and ob- 

23 



354 



ELEMENTS OF ANALYTICAL MECHANICS. 




viously expresses the value of adhesion on each unit of surface, for 
making 

-S=l, 
we have 

A := a. 

To find the adhesion between solids and liquids, suspend the solid 
from the balance, with its polished surface downward and in a hori- 
zontal position ; note the weight of the solid, 
then bring it in contact with the horizontal 
surface of the fluid and note the indication of 
the balance when the separation takes place, 
on drawing the balance up ; the difference be- 
tween this indication and that of the weight 
will give the adhesion ; and this divided by 
the extent of surface, will give, as before, the 
co-efficient a. But in this experiment two 
opposite conditions must be carefully noted, 
else the cohesion of the elements of the liquid 
for each other may be mistaken for the adhe- 
sion of the solid for the fluid. If the solid 
on being removed take with it a layer of the 
fluid ; in other words, if the solid has been 

wet by the fluid, then the attraction of the elements of the solid 
for those of the liquid is stronger than that of the elements of the 
liquid for each other, and a will be the unit of adhesion of two 
surfaces of the fluid. If, on the contrary, the solid on leaving the 
fluid be perfectly dry, the elements of the fluid will attract each 
other more powerfully than they will those of the solid, and a will 
denote the unit of adhesion of the solid for the liquid. 

It is easy to multiply instances of this diversity in the action of 
solids and fluids upon each other. A drop of water or spirits of 
wine, placed upon a wooden table or piece of glass, loses its globu- 
lar form and spreads itself over the surface of the solid ; a drop of 
mercury will not do so. Immerse the finger in water, it becomes 
wet; in quicksilver, it remains dry. A tallow candle, or a feather 




APPLICATIONS. 



355 



from any species of water-fowl, remains dry though dipped in water. 
Gold, silver, tin, lead, &c., become moist on being immersed iu 
quicksilver, but iron and platinum do not. Quicksilver when poured 
into a gauze bag will not run through ; water will : place the gauze 
containing the quicksilver in contact with water, and the metal will 
also flow through. 

It is difficult to ascertain the precise value of the force of adhe- 
sion between the rubbing surfaces of machinery, apart from that of 
friction. But this is attended with little practical inconvenience, as 
long as a machine is in motion. The experiments of which the 
results are given in Tables VI, Yll and VIII, and which are applicable 
to machinery, were made under considerable pressures, such as those 
with which the parts of the larger machines are accustomed to move 
upon one another. Under such pressures, the adhesion of unguents 
to the surfaces of contact, and the opposition to motion presented 
by their viscosity, are causes whose influence may be safely disre 
garded as compared with that of friction. In the cases of lighter 
machinery, however, such as w^atches, clocks, and the like, these 
considerations rise into importance, and cannot be neglected. 

STIFFNESS OF CORDAGE. 



§ 310. — Conceive a wheel turning 
freely about an axle or trunnion, and 
having in its circumference a groove to 
receive a cord or rope. A weight W, 
being suspended from one end of the 
rope, while a force F, is applied to the 
other extremity to draw it up, the 
latter will experience a resistance in 
consequence of the rigidity of the rope, 
which opposes every effort to bend it 
around the wheel. This resistance must, 
of necessity, consume a portion of the 
work of the force F. ITie measure of 
the resistance due to the rigidity of cordage has been made the 




356 ELEMENTS OF AN"ALyTICAL MECHANICS. 

subject of experiment by Coulomb y ,^ncl, according to him, it 
results that for the same cord an^: same wheel, this measure is 
composed of two parts, of which one remains constant, while the 
other varies with the weight W, and is directly proportional to it; 
so that, designating the constant part by IT, and the ratio of the 
variable part to the weight W by /, the measure will be given by 
the expression 

K+ I. W; 

in which IC represents the stiffness arising from the natural torsion 
or tension of the threads, and / the stiffness of the same cord due to 
a tension resulting from one unit of weight ; for, making W =: 1, the 
above becomes 

K-\- I. 

Coulomb also found that on changing the -^heel, the stiffness varied 
in the inverse ratio of its diameter ; so that if 

K+ I. W 

be the measure of the stiffness for a w^heel of one foot diameter, then 

will 

K -\- I. W 

be the measure when the wheel has a diameter of 2 i?. A table 
giving the values of K and / for all ropes and cords employed in 
practice, when wound around a wheel of one foot diameter, and sub- 
jected to a tension arising from a unit of weight, would, therefore, 
enable us to find the stiffness answering to any other wheel and 
weight whatever. 

But as it w^ould be impossible to anticipate all the different sizes 
of ropes used under the various circumstances of practice. Coulomb 
also ascertained the law^ which connects the stiffness with the diame- 
ter of the cross-section of the rope. To express this law in all cases, 
he found it necessary to distinguish, 1st, neio white rope, either dry 
or moist ; 2d, white ropes parti?/ worn, either dry or moist ; 3d, tarred 
ropes ; 4th, packthread. The stiffness of the first class he found nearly 
proportional to the square of the diameter of the cross-section ; that 



APPLICATIONS. 357 

of the second, to the square root of the cube of this diameter, nearly ; 
that of the third, to the number of yarns in the rope ; and that of 
the fourth, to the diameter of the cross-section. So that, if S denote 
the resistance due to the stiffness of any given rope ; d the ratio of 
its diameter to that of the table ; and n the ratio of the number of 
yarns in any tarred rope to that of the table, we shall have for 

JVew white rope, di-ij or moist. 

^= -^^-- ••■•••• (-) 

Half worn white'ropi, dry or moist. 
Tarred rope. 
' Packthread. 

S = ..^±^.. ..... (5X4) 

For packthread, it will always be sufficient to use the tabular 
values given, corresponding to the least tabular diameters, and substi- 
tute them in Equation (514). An example or two will be sufficient 
to illustrate the use of these tables. 

JExample \st. Eequired the resistance due to the stiffness of a new 
dry white rope, whose diameter is 1,18 inches, when loaded with 
a weight of 882 pounds, and wound about a wheel 1,64 feet in 
diameter. 

Seek in No. 1, Table X, the diameter nearest that of the 'given 
rope ; it is 0,79 ; hence, 

1.18 
0^ 

and from the table at the side, 

cZ2 = 2,25. 
From No. 1, opposite 0,79, we find 

K= 1,6097, 

/ = 0,03195; 



d = j^-^^ = 1,5 nearly; 



358 ELEMENTS OF ANALYTICAL MECHANICS. 

ft. 
which, together with the weight W = 882 lbs., and 2 i2 = 1,64, 

substituted in Equation (511), give 

S = 2,25 . W + ^03^X882 ^ ^ -^^^ 

which is the true resistance due to the stiffness of the rope in 
question. 

Example 2d. What is the resistance due to the stiffness of a 
white rope, half worn and moistened with water, having a diam- 
eter equal to 1.97 inches, wound about a wheel 0,82 of a foot in 
diameter, and loaded with a weight of 2205 pounds'? 

The tabular diameter in No. 4, Table X, next less than 1,97, 
is 1,57, and hence, 

d= —^ = 1,3 nearly; 
the square root of the cube of which is, hj the table at the side, 

d^ = 1,482. 
In No. 4 we find, opposite 1,57, 

K = 6,4324, 

/ = 0,06387 ; 

ft. 
which values, together with W = 2205 lbs., and 2 i2 = 0,82, in 

Equation (512), give 

lbs. lbs. 

S = 1,482 X M324 4- 06387 x 2205 ^ ^^ »J^^_ 

which is the required resistance. 

Example 3c?. What is the resistance due to the stiffness of a 
tarred rope of 22 yarns, when subjected to the action of a weight 
equal to 4212 pounds, and wound about a wheel 1,3 feet diameter, 
the weight of one running foot of the rope being about 0,6 of a 
pound ? 

By referring to No. 5, Table X, we find the tabular number of 
yarns next less than 22 to be 15, and hence, 

22 

n = — = 1,466 nearly. 

xO 



APPLICATIONS. 



359 



In the same table, opposite 15, we find 

K = 0,7664, 
/ = 0,019879; 

which, together with TT = 4212, and 2 i2 = 
give 

,S' = 1,466 



1,3, in Equation (513), 
0,7664 + 0,019879 x 4212 



1,3 



lbs. 

95,188. 



Example ^th. Required the resistance due to the stiffness of a 
new white packthread, whose diameter is 0,196 inches, when moist- 
ened or wet with water, wound about a wheel 0,5 of a foot in 
diameter, and loaded with a weight of 275 pounds. 

The lowest tabular diameter is 0,39 of an inch, and hence 

^ 0,196 

^ = o;39o = ^'^ ^^"'^^- 

In No. 2, Table X, we find, opposite 0,39, 

K =z 0,8048, 
/ = 0,00798 ; 

0,5, we find, after substituting in 



which, with W = 275, and 2B 
Equation (514), 



S = 0,5 



0,8048 + 0,00798 x 275 
0,5 



= 2,999. 



§311. — The resistance just found 
is expressed in pounds, and is the 
amount of weight which would be 
necessary to bend any given rope 
around a vertical wheel, so that 
the portion A U, between the first 
point of contact A, and the point 
^, where the rope is attached to 
the weight, shall be perfectly straight. 
The entire process of bending takes 
place at this first or tangential 
point A ] for, if motion be com- 




360 ELEMENTS OF ANALYTICAL MECHANICS. 

municated to the wheel in the direction indicated by the arrow* 
head, the rope, supposed not to slide, will, at this point, take and 
retain the constant curvature of the wheel, till it passes from the 
latter on the side of the power F, When, therefore, by the motion 
of the wheel, the point m of the rope, now at the tangential point, 
passes to m\ the working point of the force S will have described 
in its own direction the distance A D. Denoting the arc described 
by a point at the unit's distance from the centre of the wheel 
by s^ , and the radius of the wheel by i?, we shall have 

AD — Rs^', 

and representing the quantity of work of the force aS' by L^ we get 

replacing S by its value in Equations (511) to (514), 

K + I .W 
L^Rsrd, ^ (515) 

3 

in which d^ represents the quantity c?^^ ^2^ ^^ q^ ^^ jn Equations (511) 
to (514), according to the nature of the rope. 

Example. — Taking the 2d example of § 310, and supposing a por- 
tion of the rope, equal to 20 feet in length, to have been brought 
in contact with the wheel, after the motion begins, we shall have 

X = 20 X 266,109 = 5322,18 units of work; 

that is, the quantity of work consumed by the resistance due to 
the stiffness of the rope, while the latter is moving over a distance 
of 20 feet, would be sufficient to raise a weight of 5322.18 pounds 
through a vertical height of one foot. 



FRICTION ON PIVOTS, AND TRUNNIONS. 

§312. — All rotating pieces, such as wheels supported upon other 
pieces, give rise by their motion to friction. This is an important 
element in all computations relating to the performance of machinery. 
It seems to be different according as the rotating pieces are kept 



APPLICATIONS. 



3G1 



in place hy trunnions or by 
pivots. By trunnions are meant 
cylindrical projections a a from 
the ends of the arbor ^ ^ of a 
wheel. The trunnions rest on the 
concave surfaces of cylindrical 
boxes C7-Z), with which they usu- 
ally have a small surface of 
contact 7?z, the linear elements 
of both being parallel. Pivots 
are shaped like the trunnions, 
but support the weight of the 
wheel and its arbor upon their 
circular end, which rests against 
the bottom of cylindrical sock- 
ets FGHI. 



PIVOTS. 



Let iV denote the force, in the direction of the axis, by which 
the pivot is pressed against the 



bottom of the socket. This force 
may be regarded as passing 



through the centre of the cir- 
cular end of the pivot, and as 
the resultant of the partial pres- 
sures exerted upon all the ele- 
mentary surfaces of which this 
circle is composed. Denote by 
A the area of the entire circle, 
then will the pressure sustained 
by each unit of surface be 

N 

A ' 

and the pressure on any small portion of the surface denoted by a, 

will obviously be 

a. iV^ 




362 



ELEMENTS OF ANALYTICAL MECHANICS. 



and the friction on the same will be 

f.a.N 



This friction may be regarded as applied to the centre of the ele- 
mentary surface a ; it is opposed to the motion, and the direction of 
its action is tangent to the circle described by the centre of the 
element. Denote the radius of this circle by x^ then will the mo- 
ment of the friction be 

. a.N 

Now, if ^ denote the length of any variable portion of the circumfer- 
ence at the unit's distance from the centre (7, then will 

a =i X . d^ , dx\ 
also, 

which substituted above give 

f-N 

and by integration, 



2 . dx . dz 



f'N 



/R /»2 ff 

x^ dx I d $ 



'TtB^ 



= /.iVr.|i2; 



(516) 



whence we conclude, that, in the fric- 
tion of a pivot, we may regard the 
whole friction due to the pressure as 
acting in a single point, and at a dis- 
tance from the centre of motion equal 
to two-thirds of the radius of the base 
of the pivot. This distance is called 
the mean lever of friction. 

§ 313. — If the extremity of the pivot, 
instead of rubbing upon an entire circle, 
is only in contact with a ring or sur- 
face comprised between two concentric 




APPLICATIONS. 363 

circles, as when the arbor of a wheel is urged in the direction of 
its length by the force iV" against a shoulder deb a-, then will 

and the integration will give 

/R /»2 T 

x^ dx ds _- „,, 

in which R denotes the radius of the larger, and R' that of the 
smaller circle. 

Finally, denote by I the breadth of the ring, that is, the dis- 
tance A' A ; by r, its mean radius or distance from C to a point 
half way between A' and A^ and we shall have 

R = r + i /, 

R'^T-\l', 

substituting these values above and reducing, we have 

Z2- 



/.iVTx ^r + .v—]; (517) 



and makinsT 






we obtain, for the moment of the friction on the entire ring, 

f-N.r (518) 

The quantity r^ is called the mean lever of friction for a ring. Since 
the whole friction fN may be considered as applied at a point 

/2 

whose distance from the centre is j R^ or r^ = r + T77-' according 
as the friction is exerted over an entire circle or over a ring, 
and since the path described by this point lies always in the di- 
rection in which the friction acts, the quantity of work consumed 
by it will be equal to the product of its intensity fN into this 
path. Designating the length of the arc described at the unit's 
distance from C by s^ , the path in question will be cither 

J i2 5, , or r, s, ; 



S64: ELEMENTS OF ANALYTICAL MECHANICS, 

and the quantity of work either 

for an entire circle, or 

for a ring. Let Q denote the quantity of work consumed by fric- 
tion in the unit of time, and n the number of revolutions performed 
by the pivot in the same time ; then will 

s^ =: 2 -Tr X n ; 
and we shall have 

Q =iir.B.f.JV.n ...... (519) 

for the circle, and 

Q=:2^'f'JV. (r + —^ .n .... (520) 

for a ring; in which 'jf = 3,1416. 

The co-efficient of friction /, when employed in either of the fore- 
going cases, must be taken from Table VI, VII, or VIII. 

Exam.'ple. — Required the moment of the friction on a pivot of 
cast iron, working into a socket of brass, and which supports a 
weight, of 1784 pounds, the diameter of the circular end of the 
pivot being 6 inches. Here 

in. ft. 

i2 =r f =: 3 = 0,25, 

lbs. 

N = 1784, 
/ = 0,147 ; 
which, substituted in Equation (516), gives 

lbs. ft. 

0,147 X 1784 X f X 0,25 = 43,708. 

And to obtain the quantity of work in one unit of time, say a 
minute, there being 20 revolutions in this unit, we make n = 20, 
and If — 3,1416 in Equation (519), and find 

Q ^ ^ X 3,1416 X 0,25 X 0,147 X 1784 X 20 =: 5492,80; 



APPLICATION'S. 365 

that is to saj, during each unit of time, there is a quantity of 
work lost which would be sufficient to raise a weight of 5492,80 
pounds through a vertical distance of one foot. 

Example. — Required the moment of friction, when the pivot sup- 
ports a weight of 2046 pounds, and works upon a shoulder whose 
exterior and interior diameters are respectively 6 and 4 inches ; the 
pivot and socket being of cast iron, with water interposed. 

I = — - — = 1 inch, 

r = 2 + 0,5 = 2,5 inches, 

(1)2 in. ft. 

^' = ^'^ + 12X2,5 = ^'^^^^ = ^'^^^^' 
iV^ = 2046 pounds, 
/= 0,314; 

which, substituted in Expression (518), gives for the moment of friction, 

lbs. ft. 

0,314 X 2046 X 0,2111 = 135,62. 

The quantity of work consumed in one minute, there being sup 
posed 10 revolutions in that unit, will be found by making in 
Equation (520), -tt = 3,1416 and n == 10, 

^ = 2 X 3,1416 X 0,314 x 2046 X 0,211 X 10 = 8517,24; 

that is to say, friction will, in one unit of time, consume a quantity 
of work which would raise 8517,24 pounds through a vertical dis- 
tance of one foot. The quantity of work consumed in any given 
time would result from multiplying the work above found, by the 
time reduced to mmutes. 

TRUNNIONS. 

§314. — The friction on trunnions and axles, which we now pro- 
ceed to consider, gives a considerably less co-efficient than that which 
accompanies the kinds of motion referred to in § 308. This will 
appear from Table IX, which is the result of careful experiment. 

The contact of the trunnion with its box is along a linear cle- 



see 



ELEMENTS OF ANALYTICAL MECHANICS. 




ment, common to the surfaces of both. A section perpendicular to 
its length would cut from the trunnion and its box, two circles tan. 
gent to each other internally. The trunnion being acted on only by 
its weight, would, when at rest, give this tangential point at o, the 
lowest point of the section ^ o g^ of the box. If the trunnion be put 
in motion by the application of a force, it would turn around the 
point of contact and roll 
indefinitely along the sur- 
face of the box, if the 
latter were level ; but this 
not being the case, it will 
ascend along the inclined 
surface op to some point 
as m, where the inclina- 
tion of the tangent umv 
is such, that the friction 
is just sufficient to pre- 
vent the trunnion from sliding. Here let the trunnion be in equili- 
brio. But the equilibrium requires that the resultant of all the 
forces which act, friction included, shall pass through the point m 
and be normal to the surface of the trunnion at that point. The 
friction is applied at the point m ; hence the resultant iV of all the 
other forces must pass through m in some direction as md^ the 
friction acts in the direction of the tangent; and hence, in order 
that the resultant of the friction and the force iV shall be normal to 
the surface, the tangential component of the latter must, when the 
other component is normal, be equal and directly opposed to the 
friction. 

Take upon the direction of the force JV" the distance md to 
represent its intensity, and form the rectangle ad bm, of which 
the side m b shall coincide with the tangent, then, denoting the 
angle dma by 9, will the component of iV perpendicular to the tan- 
gent be 

i\r . cos 9 ; 

and the friction due to this pressure will be 

/. iV. cos (p. 



APPLICATIONS, 



36T 



The component of iV^, in the direction of the tangent, will be 

N . sin 9 ; 
and as this must be equal to the friction, we have 

/. iV. C0S9 = iV^. sin 9 J (521) 

whence, 

/ = tan 9 ; 

that is to say, the ratio of the friction to the pressure on the trun- 
nion is equal to the tangent of the angle which the direction of the 
resultant iV, of all the forces except the friction^ makes with the nor- 
mal to the surface of the trunnion at 
the point of contact. This gives an easy 
method of iinding the point of con- 
tact. For this purpose, we have but 
to draw through the centre A a line 
A Z, parallel to the direction of iV, 
and through A, the line Am, making 
with A Z an angle of which the tan- 
gent is /; the point m, in which this 
line cuts the circular section of the / 

n 

trunnion, will be the point of contact. 

Because madb, last figure, is a rectangle, we have 

iV^2 _ iV^2cos29 4- iV2sin29; 

and, substituting for N"^ sin^ 9 its equal /^ N^ cos^ 9, we have 

iV2 _ iV^2cos29 -f/2iV^2cos2 9 - N^ COs"^ (p (1 +/2); 

whence, 

1 
iVcos 9 — iV X 




VTTT' 



and multiplying both members by /, 

/ . iV . cos 9 = iV . 



/ 



(522) 



but the first member is the total friction ; whence we conclude 
that to find the friction upon a trunnion^ we have hut to multiply the 



368 



ELEMENTS OF ANALYTICAL MECHANICS. 



resultant of the forces which act upon it by the unit of friction^ found 
in Table ZZ", and divide this product by the square root of the square 
of this same unit increased hy unity. 

This friction acting at the extremity of the radius R of the trun- 
nion and in the direction of the tangent, its moment will be 

/ 



N 



-/I +/ 



X B. 



(523) 



And the path described by the point of application of the friction 
being denoted by Rs^^ the quantity of work of the friction will be 



/ 



JSf.R.s. X ^ . , 



(524) 



in which s^ denotes the path described by a point at the unit's dis- 
tance from the centre of the trunnion. Denoting, as in the case of 
the pivot, the number of revolutions performed by the trunnion in 
a unit of time, say a minute, by n ; the quantity of work performed 
by friction in this time by Q^ ; and making -r = 3,1416, we have 



and 



Sj = 2'jf .n-, 



Q^ =2ii ,R.n.N. 



f 



V^+P 



(525) 



When the trunnion remains fixed and does not form part of the 
rotating body, the latter will turn about the trunnion, w^hich now 
becomes an axle, having the centre of 
motion at A^ the centre of the eye of 
the wheel ; in this case, the lever of fric- 
tion becomes the radius oi the eye of 
the wheel. As the quantity of work 
consumed by friction is the greater, 
Equation (525), in proportion as this 
radius is greater, and as the radius of 
the eye of the wheel must be greater 

than that of the axle, the trunnion has the advantage, in this respect 
over the axle. 




APPLICATION-S. 



369 



The value of the quantity of work consumed by friction is wholly 
independent of the length of the trunnion or axle, and no advantage 
is therefore gained by making it shorter or longer. 



THE CORD. 



§ 315. — The cord and its properties have been considered in part 
at § 58. It is now proposed to discuss its action under the opera- 
tion of forces applied to it in any manner whatever. 

Let the points A\ A", A'", be connected with each other by 
means of two perfectly flex- 
ible and inextensible cords 
A' A", A" A'", the first 
point being acted upon by 
the forces P', P'\ &c. ; the 
second by the forces Q', Q", 
&c. ; and the third by the 
forces «S", S'\ &c. ; and sup- 
pose these forces to be in 
equilibrio. Denote the co- 
ordinates of A' by x' y' z\ 
^"by a;"/'2", and^"'by 




x'" y'" z'". Also, the alge- 



braic sum of the components of the forces acting at A' in the direc- 
tion of xyz, by X' T Z\ at A" by X" Y" Z", and at A'" by 
X'" Y'" Z'". Then will, § 101, 



X' 5x' + F' 5y' + Z' Sz' ^ 
+ X" 8 x" + Y" 8 y" + Z" 8z" \=0. . . 
+ X" dx'" + Y'" 8y"' + Z"'8z'" . 

Denote the length A' A" by/, and A" A'" by ^; then will 



(526) 



n = g- ^" - x"f + {y'" - y"f + (z"' - z"f ^ 0. 



(527) 



The displacement by which we obtain the virtual velocities whose 

24 



370 ELEMENTS OF ANALYTICAL MECHANICS. 

projections are Sx'^ 5 y\ Sz', &c., is not wholly arbitrary; but must 
be made so as to satisfy the condition 

Sf= and §g = 0. (528) 

Differentiating Equations (527), and writing for dx', dy\ dz\ 
^ x\ ^y\ d z\ &c., we find 

{x'' - x'){Sx" ^ 8x^) + if' - y'){5y'' - S/) + (^" - ^'W - Sz') _ ^ 

/ 
(a,''f-x'ry^(^Sx'"-§x") + (y'''-/'){dy'"-dy")-^{z'''-z'')(§z'''-Sz" _ 

9 ~ 

These being multiplied respectively by X' and X'", and added to 
Equation (526), we obtain by reduction, and by the principle of 
indeterminate co-efficients, exactly as in §213, 

r" — r' 



Y' - X 



/ 

, y-y' 



f 



Z' — X' 



/ 



(529) 



X^' + V . 



x'' _ x' x'" - x'' 
— X • 



/ 



0; 



Y" + X 
Z" + X' 



, y - y 



f 

z" - z' 



v''y"-y" = ,- 



f 

X'" + V" 



= 0; 



(530) 



9 

.in „Jt 



0; 



Y"' + \"' . y- — ^ - 



,ni „tt 



Z"' + X'" 



0; 



(531) 



Taking from each group its first equation and adding, and doing 
the same for the second and third, we have 



X' + X" + X'" = ; 
Y' + Y" + F'" = ; 
Z' + Z" + Z'" = 0. J 



(532) 



APPLICATIONS. 



871 



That is, the conditions of equilibrium of the forces are, §80, the 
same as though they had been applied to a single point. 

To find the position of the points, eliminate the factors X' and 
X'", and for this purpose add the first, second and third equations 
of group (530) to the corresponding equations of group (531), and 
there will result 



X" + X'" + -^ {x" - x' 



0; 



T' -{■ T" ^ J {y" - y') = 0; 
Z" + Z'" + ~ (2" - z') = 0. 



from which we find by elimination, 



T77/ I p-/// 



y" - y 



{X" + X") = ; 



Z'^ + Z"' - -r, {X" + X") = 0. 



From group (529), by eliminating X', 



(533) 



y, _ y^^ — / x = 0', 

X — X 



Z' - 



-,X' = ^; 



(534) 



and finally from group (531) we obtain, by eliminating X"', 



y/// 



Z'" — 



X — X 



'11 ^11 



0; 



X' 



(535) 



Equations (532), (533), (534) and (535), involve all the conditions 
necessary to the equilibrium, and the last three groups, in connection 
with group (527), determine the positions of the points A'^ A" 
and A"\ in space. 



31G. — The reactions in the system which impose conditions oa 



872 ELEMENTS OF ANALYTICAL MECHANICS. 

the displacement will be made known by E(][uation (331), which, 
because 

ld(x"' -x")j ld(y'"-y'')j ld{z''' - s'')j "" ' 
becomes for the cord A' A", 

V = iV' ; 
and for the cord A" A"\ 

V" = N"' ; 

from which we conclude, that X' and X"' are respectively the ten- 
sions of the cords A' A" and A" A'". 

This is also manifest from Equations (529) and (531); for, by 
transposing, squaring, adding and reducing by the relations, 

p - 1. 

we have 



(536) 



in which JS' and R'" are the resultants of the forces acting upon 
the points A' and A'" respectively. 

Substituting these values in Equations (529) and (531), we have 

^ _ x" - x' ^ Y^ _ y" - y' ^ Z^ _ z'' - z' ^ 
R' ~~ f ' R' ~ f ' R' ~ f ' 

X'" x'" — x" T" y'" — y' 



R'" ~" g ' R'" g ' R'" g ' 

whence the resultants of the forces applied at the points A' and A'"^ 
act in the directions of the cords connecting these points with the 
point A!\ and will be equal to, indeed determine the tensions of 
these cords. 



APPLICATIONS. 373 

§317. — From Equations (532), we have by transposition, 

X'^ = - (X'" + X') ; Y" = - {¥"' + y) ; Z" = - {Z'" + Z'). 

Squaring, adding and denoting the resultant of the forces applied 
at A" by R'\ we have 



R" = -v/(Z'" 4- X'f + {F'" + F'}'' + {Z'" + Z'y ' . (537) 
and dividing each of the above equations by this one 

X'" + X' 



X" 
R" 

Y" 
R" 

Z" 

R" 





R" 




Y" + 


Y' 




R" 




Z' 


' + 


Z' 



(538) 



whence, Equation (537), the resultant of the forces applied at A" is 
equal and immediately opposed to the resultant of all the forces 
applied both at A' and A'" 

If, therefore, from the point 
A'\ distances A" m and A" n 
be taken proportional to R' and 
R'" respectively, and a paral- 
lelogram A" m Cnhe constructed, 
A" C will represent the value of 
R". If A' A" A'" be a contin- 
uous cord, and the point A" 
capable of sliding thereon, the 
tension of the cord would be 
the same throughout, in which 
case i2' would be equal to R"\ 
and the direction of R" would 
bisect the angle A' A" A'". 

The same result is shown if, 
instead of making J/ — and 
6 ff = separately, we make 





374 ELEMENTS OF ANALYTICAL MECHANICS. 

^ (/ -f- ^) = 0, multiply by a single indeterminate quantity X, 
and proceed as before. 

§ 318.— Had there been four 
points, A', A'', A"' and A''', 
eonnected by the same means, 
the general equation of equili- 
brium M^ould become, by call- 
ing A the distance between the 
pomts, A'" and A^^, 

X' S x' + X'^ S x'' + X^' 6 x"' + Xi^ ^ x^^ 
4- Y' dy' -\- Y" 5 y" -f T" § y'" -f Y^^ S y^^ 
+ Z' S z' + Z'' S z" -f Z'" S z'" -f Zi^ ^ si^ 

and from which, by substituting the values of ^ f^ ^g and ^ A, the 
following equations will result, viz.: 




0; 



X' - \' 



f 



Y'-^vJ -y 



f 



Z' -V ' 



f 



= 0, 



0, 



= 0, 



(539) 



X' + X' 

Y" + X 
Z" + X 



/ 

, y" — y' 



y'" - y" 



f 



= 0, 



/ 



(540) 



X'' -f X" . 



r^' + X' 



y'" - y' 



= 0, 



y . (541) 



Z'" + X" . ~ X'" . p— - = 0, 



APPLICATIONS, 



375 



X'^ + V 



x^^ - x'" 



o,iy _ y'" 

Ziy + \"' = 0, 



(542) 



Eliminating the indeterminate quantities Xj X" and X'", we obtain 
nine equations, from which, and the three equations of conditions 
expressive of the lengths of/, ff, and h, the position of the points A', 
A", A"\ and A^^ may be determined* 

If there be n points, connected in the same way and acted apon 
by any forces, the law which is manifest in the formation of Equa- 
tions (539), (540), (541), and (542), plamly indicates the following 
n equations of equilibrium : 



X'- — X' . 



/ 



0, 



F - X' . y^^ = 0, 

x" — z' 
Z' - X' . ^-^ = 0, 



(543) 



X" + X' 



:A - X" . ~ = 0, 



F" + X' . 



, y"-y' ,. y'"-y" 



f 



\" . 



-0, 



Z" + X' 


z — z 


f 


X'" -f X" 


x'" - x" 


9 


F'" + X" 


y'" - y" 
9 


Z'" 4- X" . 


^>" y" 



z — z 
X" . = 0, 



x-.^ 



.nr y'^ - y" 



= 0, 
= 0, 

= 0, 



(544) 



(545) 



376 ELEMENTS OF ANALYTICAL MECHANICS 



^n-i + K 



r^-i + K. 



Zn.l + K. 



-2 



-K-^ 



•2 • 



yn-1 — Vn-i 



- X, 



I 

I 

z„ — z 



I 



= 0, 



l^^ = 0, 



= 0, 



(546) 



^n + \-l 



I 



0, 



F„ + x„„,.^^^^^ = o, 



Zr. +X,„i 






0. 



(547) 



In which X, with its particular accent, denotes the tension of the 
cord into the difference of whose extreme co-ordinates it is multi- 
plied. 

Adding together the equations containing the components of the 
forces parallel to the same axis, there will result 



X' + X" + X'" + x^^ 
Y' -4- Y" + Y'" + 1^^"^ 



z^ = 0, 



(548) 



from which we infer, that the conditions of equilibrium are the 
same as though the forces were all applied to a single point. 

From group (543), we find by transposing, squaring, adding and 
extracting square root, 



and dividing each of the equations found after transposing in group 
(543) by this one, 



X' 


= 


rr" 


— 


x' 




B' 




/ 




> 


Y' 


= 


yl 


T 


iL 


5 


Z' 


= 


z" 


— 


z' 




R' 




f 







APPLICATIOXS. 



377 



Treating the equations of group (547) in the same way, we 
have 





— 


— 


x„ 




-1 




— 


— 


yn_ 


I 


^ 




= 


— 


^n 


l 


-J^ 




whence, the resultants 

of the forces applied 

to the extreme points 

A' and A^ , act in the 

direction of the extreme cords. And from Equations (548) it appears 

that the resultant of these two resultants is equal and contrary to 

that of all the forces applied to the other points. 



§319.— If the extreme points be fixed, X', Y', Z' and X,, F„, Z„, 
will be the components of the resistances of these points in the 
directions of the axes ; these resistances will be equal to the ten- 
sions X' and X„ of the cords which terminate in them. Taking the 
sum of the equations in groups (543) to (547), stopping at the point 
whose co-ordinates are ar„_^, y^;„, 0„_„, we have 



X' + 2 X - X, 

T' + 2 r - x„ 

^' + 2 Z - X„ 



^n-m — ^n 



= 0; 



= 



Z — 2 1 



(549) 



in which 2 X, 2 F", 2 Z, denote the algebraic sums of the components 
in the directions of the axes of the active forces ; X„_^ the tension 
on the side of which the extreme co-ordinates are a:„_„, y„_;„, 2f,_m, 
and ;r„_„_i, 2/„-m_T, ^n-m-i] and l^^ the length of this side. 

§320. — Now, suppose the length of the sides diminished and 



878 



ELEMENTS OF ANALYTICAL MECHANICS. 



their number increased indefinitely ; the polygon will become a 
curve ; also, making X^^ = t^ we have 

k^m = ds, 

s being any length of the curve ; and Equations (549) become 

as 



r + 2F- t-^ = 0: 

ds 



d s 



0; 



(550) 



which will give the curved locus of a rope or chain, fastened at 
its ends, and acted upon by any forces whatever, as its own weight, 
the weight of other materials, the pressure of winds, currents of 
water, &c., &c. 

This arrangement of several points, connected by means of flexi- 
ble cords, and subjected to the action of forces, is called a Funi- 
cular Machine. 

§321. — If the only forces acting be pressure from weights, we 
have, by taking the axis of z vertical. 



X^: = X'" = X'^^ &c. == ; Y" = Y'" &;c. 







and from Equations (543) to (547), 



X' ^X'."" /" =\' 



^n-\ 



In 



whence, the tensions on all the cords, estimated in a horizontal 
direction, are equal to one another. Moreover, we obtain from the 
same equations, by division. 



y" - y' 



X — X 






Xn 



APPLICATIONS. 



379 



These are the tangents of the angles which the projections of the 
sides on the plane xy make with the axis x. The polygon is 
therefore contained in a vertical plane. 



TEE CATENAEY. 



g322. — If a single rope or chain cable be taken, and subjected 
only to the action of its own weight, it will assume a curvilinear 
shape called the Catenary curve. It will lie in a vertical plane. 
Take the axes z and x in this plane, and z positive upwards, then 
will 

2X=0; 2F=0; Z' = 0; SZ^-TT; 

in which W denotes the weight of the cable, and Equations (550) 
become 

dx 

ds 

d z 



X' -t 



= 0, 



Z' -w 



t'- =0. 

a s 



(551) 



These are the differential equations of the curve. The origin 
may be taken at any point. 
Let it be at the bottom point 
of the curve. The curve 
being at rest, will not be 
disturbed by taking any one 
of its points fixed at pleas- 
ure. Suppose the lowest 
point for a moment to be- 
come fixed. As the curve 

is here horizontal, Z' = 0, §319, and from the second of Equations 
(551), we have 

dz 




W = 



d. 



(55-) 



whence, the vertical component of the tension at any point as of 
the curve, is equal to the weight of that part of the cable between 
this point and the lowest point. The first of Equations (551) shows 



380 ELEMENTS OF ANALYTICAL MECHANICS. 

that the horizontal component of the tension at is equal to the 
tension at the lowest point, as it should be, since the horizontal 
tensions are equal throughout. 

Taking the unit of length of the cable to give a unit of weight, 
which would give the common catenary, we have W =: s\ and, de- 
noting the tension at the lowest point by c, we have 

t - ± ^S^ -\- C2, 

and from Equation (552), 

s ' ds 
dz =1 zp — ______ 

-V/C2 + 6-2 

Taking the positive sign, because z and s increase together, inte- 
grating, and finding the constant of integration such that when 
2 rr 0, we have s = 0, 



2 + c =1 y c2 + §2 . 
whence, 

§2 — ^2 _|_ 2c0. 

Also, dividing the first of Equations (551) by Equation (552), 

dx c c 

dz~ s ~ ^^2_^2cz ' 

and integrating, and taking the constant such that x and z vanish 
together. 



, z + c -\r Vs2 + 2 c z 

rr = c • log • • • (553) 

c 

which is the equation of the catenary. 

This equation may be put under another form. For we may 
write the above. 



c e c = z + c -{- y^z + c) 2 — 
transposing z -\- c and squaring, 



c2 . gc — 2c e' {z -{■ c) 



whence, 

2 + c = I c • (e«" + e~ «"). (554) 



APPLICATIONS. 



381 



Also, 

and by substitution, 



s = ■\/{z + c)2 — c2, 



s = ic'{e~ ^ e~c). (555) 

§323. — If the length of the portion of the cable which gives a 
unit of weight were to vary, the variation might be made such as 
to cause the area of the cross section to be proportional to the 
tension at the point where the section is made. The general Equa- 
tions (551) will give the solution for every possible case. 



FEICTION BETWEEN COEDS ANT) CYLINDRICAL SOLIDS. 

§ 324. — When a cord is wrapped around a solid cylinder, and 
motion is communicated by applying the power F at one end 
while a resistance W acts at the other, a pressure is exerted by 
the cord upon the cylinder ; this pressure produces friction, and this 
acts as a resistance. To estimate its amount, denote the radius 
of the cylinder by B, the arc of contact by 5, the tension of the 
cord at any point by t. 

The tension i being the same 
throughout the length ds = at^ 
of the cord, this element will be 
pressed against the cylinder by 
two forces each equal to t, and 
applied at its extremities a and t^ , 
the first acting from a towards 
TF, the second from t^ towards h'. 
Denoting by d the angle ab t^^ 
and by p the resultant 6 m of 
these forces, which is obviously 
the pressure oi ds against the cylinder, we have, Equation (5G), 

P 




but 



= V^M-^M^ t,tcos& = t v'2(l + cos<)) ; 
1 -}- cos^ =r 2cos2J^; (180° — &) 



ds 



382 ELEMENTS OF AITALYTICAL MECHANICS. 

and taking the arc for its sine, because 180° — d is very small^ 
■we have 

ds 

and hence, § 308, the friction on c? s will be 

The element t^ 4 of the cord which next succeeds a t^ , will have 
its tension increased by this friction before the latter can be over- 
come ; this friction is therefore the differential of the tension, being 
the difference of the tensions of two consecutive elements j whence, 

■dt=f.t--; 
dividing by t and integrating, 



log^=/.- + log(7, 



t = Ce^ (556) 



making s = 0, we have t = W = C \ whence, 



t=W'e'^', (557) 

and making s = S = at^^f^t^, we have t = F; and 

F= W-e^ (558) 

Suppose, for example, the cord to be wound around the cylinder 
three times, and f = ^ ; then will 

S =3'jf.2E = 6. 3,1416. i2 = 18,849 i?, 
and 

or, 

F= TF. 535,3; 

that is to say, one man at the end W co^^ld resist the combined 
effort of 535 men, of the same strength as himself, to put the cord 
in motion when w^ound three times round the cylinder. 



APPLtCATIONS. 



383 



THE INCLIXED PLANE. 

§ 325. — The inclined plane is used to support, in part, the weight 
of a body while at rest or in motion upon its surface. 

Suppose a body to rest with one of its faces on an incHned plane 
of Avhich the Equation is 

L = cos ax -{- cos b y -\- cos cz — d = ^ ] • • • • {a) 
in which d denotes the distance of the plane from the origin of co- 
ordinates, and a, b, c, the angles which a normal to the plane makes 
with the axes x, y, z, respectively. 

Denote the weight of the body by W; the power by F ; the nor- 
mal pressure by N; the angles which the power makes with the 
axes X, y, z, by a^, /3^, y^, respectively; and the path described by 
the point of application of the resultant friction by s. Then, taking 
the axis z vertical and positive upwards, and supposing the force to 
produce a uniform motion of simple translation, will, Eq. (508), 



{F.o.a,+fNp) 



dS' 
dy- 



X 



+ (i^cos^,+/iV^)(J2/ 

dz 
s 



+ (i^cos7, -]-fN^-W) 
and, Equation (a). 



= 



(6) 



cos a ^ re -f cosbS y -{- coscS z =z 0. 
Multiplying this last by X, adding and proceeding as in § 213, 



dx 
F cos a,^ -\- f N —^ + ^ cos a 



F cos[3^ -\- fN-/ -j-X cos b 



dy 
d s 



Fcosy^ -\-fN 



dz 
d s 



X cos c — PF = ; 



(o) 



and, Eq. (331), 



--■/(Syn?!)Mgy 



= x. 



{d) 



Substituting the value of X in Equations (c), the first two give by 
eliminating N, 



384: ELEMENTS OF ANALYTICAL MECHANICS. 



dx 

J h cosa ., 

a s cos p^ 

r dy cos a 

/ -— + cos ^ 
as 



+ 1 = 



(e) 



{§) 



and the first and third, by eliminating iV, 

F|/[cos7^- cosa^-^J-f-cos/^cosa — cosa^ cose =z ^If-, — l-cosa) 

If there he no friction, then will /= 0, and, Eq. (e), 

^ cosa ^°^ ^/ I 1 _ n . 

cos b cos a^ 

whence, Eqs. (45) and (a), the power must he applied in a plane 
normal hoth to the inclined plane and to the horizon. 

If without disregarding friction, the power he appHed in a plane 
fulfilling the above condition, and also con- 
taining the centre of gravity, the resultant 
friction may be regarded as acting in this 
plane, and we may take it as the co- 
ordinate plane z x, in which case 

cos 6 



O;cosft .= 0; ^~^ = 0; 




and denoting the mclination of the plane to the horizon by a, and 
that of the power to the inclined plane by 9 ; 

cos a = sin a ; cos c = — cos a ; cos y^ = sin a^ ; 
dx dz 



cos 7, — cos a. 

' d s 



= — sin a^ cos a -j- cos a^ sin a = sin (a — a J = sin 9 ; 



cos y^ cos a — cos a^ cose = sin a^ sin a + cos a^ cos a = cos (a— aj^cosqj : 
which, in Eq. (^), give 



ET_ IF (sin a -|-/ cos a) 
cos 9 -|- / sin 9 



(559) 



This supposes motion to take place wp the plane ; if the power F 
be just sufficient to permit the body to move uniformly down the 
plane, then will / change its sign, and we shall have 

_, Tr(sina — /cosa) . ^^. 

cos 9 — / sm 9 ^ ' 

And the power may vary between the limits given by these two 
values without moving the body. 



APPLICATIONS. 385 

§ 326. If the power be zero, or i^ = 0, then will 

sin a — / cos a = 0, 
or 

tan a = /, 

which is the angle of friction, § 308. 

§327. — If the power act parallel to the plane, then will 9 = 0, 

and 

i^= Tr(sina ±/cosa) (561) 

the upper sign answering to the case of motion up, and the lower, 
down the plane; the difference of the two values being 

2/Trcosa. 
If / = 0, then will 

F . B C 

that is, the power is to the weight as the height of the plane is to 
its length; and there will be a gain of power. 

§ 328. — If the power be applied horizontally, then will 9 be nega- 
tive and equal to a, and we have, by including the motion in both 

directions, 

^^ Tr(sina±/cos« ) 

cos a q:: /sin a ^ ' 

the difference of the limiting values being 

2/. W 
cos^ a. — f^ sin^ a* 

If the friction be zero, ov f =z 0, then will 

F ^ BC 

That is, the power will be to the resistance as the height of the 
plane is to its base; and there may be gain or loss of power. 

§ 329. — To find under what angle the power will act to greatest 
advantage, make the denominator in Equation (559) a maximum, 
For this purpose, we have, by differentiating, # 

— sin 9 + /cos 9 = ; 
2o 



ELEMENTS OF ANALYTICAL MECHANICS. 



whence, 

tan 9 = / 

That is, the angle should be positive, and equal to that of the fric- 
tion. 

§330. — If the power act parallel to any inclined surface to move 
a body up, the elementary quantity of work of the power and resist- 
ances will give the relation. Equation (561), 

F d s = Wds sin a, -\- Wfd s cos a. 

But, denoting the whole hori- 
zontal distance passed over by 
I = A C, and the vertical height 
hjh = B (7, we have 

d s . sin a = d h, 

d s . cos a, = d I', 




A o 



whence, substituting, and integrating, and supposing the body to be 
started from reet and brought to rest again, in which case the work 
of inertia will balance itself, we have 

Fs=zWh-\-f.W.l, (563) 

in which there is no trace of the path actually passed over by the 
body. The work is that required to raise the body through a ver- 
tical height B (7, and to overcome the friction due to its weight over 
a horizontal distance A C. 

The resultant of the weight and the power must intersect tne 
inclined plane within the polygon, formed by joining the points of 
contact of the body, else the body will roll, and not slide. 



THE LEVER. 



§331. — The Lever is a solid 
bar A B, of any form, supported 
by a fixed point 0, about which 
it may freely turn, called the ful- 
crum. Sometimes it is supported 
upon trunnions, and frequently 




APPLICATIONS. 



387 



upon a knife-edge. Levers have 
been divided into three different 
classes, called orders. 

In levers of the first order, the 
power F and resistance Q are 
applied on opposite sides of the 
fulcrum 0; in levers of the second 
order, the resistance Q is applied 
to some point between the ful- 
crum and the point of appli- 
cation of the power F; and in 
the third order of levers, the 
power F is applied between the 
fulcrum and point of applica- 
tion of the resistance Q. 

The common shears furnishes 
an example of a pair of levers 
of the first order ; the nut-crackers 
of the second ; and fire-tongs of 
the third. In all orders, the con- 
ditions of equilibrium are the 
same. 

These divisions are wholly ar- 
bitrary, being founded in no dif- 
ference of principle. The relation 
of the power to the resistances, 
is the same in all. 

Let A B be a lever supported 
upon a trunnion at 0, and acted 
upon by the power P and resist- 
ance Q, applied in a plane per- 
pendicular to the axis of the trun- 
nion. Draw from the axis of the 
trunnion, the lever arms n and 
m, being the perpendicular dis- 
tances of the power and resistance 
from the axis of motion, and 










388 ELEMENTS OF ANALYTICAL MECHANICS. 

denote them respectively by Ip and l^\ also denote the resultant of 
P and Q by N^ the radius of the trunnion by r, the co-efficient of 
friction by /, and the arc described at the unit's distance from the 
axis by Sj . 
Then, 



in which & is the angle of inclination A C B of the power to the 
resistance. Then, supposing the lever to have attained a uniform 
motion, will. Equations (508) and (524), 

Omitting the common factor d s^ , and making 



f . K r 

— _ =/; m = — ; n = -z- 



we have, 



p -m Q -^ VP2 + ^2 + 2 P ^ . cos ^ 'fn = 0. 
Transposing, squaring, and solving, with respect to P, we find, 



m -^ f n (f n cos ^ ± -/l + 2 m cos ^ + m^ -/2 ^2 ginZ &) 

If the fraction n be so small as to justify the omission of every 
term into which it enters as a factor, or if the co-efficient of friction 
be sensibly zero, then would 

| = -=i ^'''^ 

That is, the power and the resistance are to each other inversely as 
the lengths of their respective lever arms. 

If the power or the resistance, or both, be applied in a plane 
oblique to the axis of the trunnion, each oblique action must be 
replaced by its components, one of which is perpendicular, and the 
other parallel to the axis of the trunnion. The perpendicular com- 
ponents must be treated as above. The parallel components will, if 



APPLICATIONS. 



389 



the friction arising from the resultant of the normal components be 
not too great, give motion to the whole body of the lever along the 
trunnion ; and if this be prevented by a shoulder, the friction upon 
this shoulder becomes an additional resistance, whose elementary 
quantity of work may be computed by means of Eq. (520) and made 
another term in Equation (564). 



WHEEL AND AXLE. 

§ 332. — This machine consists of a wheel mounted upon an arbor, 
supported at either end by a trun- 
nion resting in a box or trunnion 
bed. The plane of the wheel is at 
right angles to the arbor ; the pow- 
er P is applied to a rope wound 
round the wheel, the resistance to 
another rope wound in the opposite 
direction about the arbor, and both 
act in planes at right angles to the 
axis of motion. Let us suppose the 
arbor to be horizontal and the re- 
sistance ^ to be a weight. 

Make 
N and N' — pressures upon the trunnion boxes at A and B ; 
R rz radius of the wheel ; 
T = radius of the arbor ; 
p and p' = radii of the trunnions at A and B ; 




5i =: arc described at unit's distance from axis of motion. 
Then, the system being retained by a fixed axis, we have 

P^IJ = PRds,; 
Q q — Q r d Si. 
The elementary work of the friction will, Eq. (524), be 

fi^yp + ^'nds,; 



390 ELEMENTS OF ANALYTICAL MECHANICS. 

and the elementary work of the stiffness of cordage, Equation 
(515), 

. K-\-I ,q . 
d, — r-dsr, 

and when the machine is moving miiformly, 

FEds,-Qrdsi-f{J}fp-{-]V'p')ds,-d^'^^f-^.r'ds, = 0', • (567) 

The pressures iV and N' arise from the action of the power P, the 
weight of the machine, and the reaction of the resistance Q, in- 
creased by the stiffness of cordage. To find their values, resolve 
each of these forces into two parallel components acting in planes 
which are perpendicular to the axis of the arbor at the trunnion 
beds; then resolve each of these components which are oblique to 
the components of Q into two others, one parallel and the other 
perpendicular to the direction of Q. 

Make 
w = weight of the wheel and axle, 
ff = the distance of its centre of gravity from A, 
p = the distance m A, 

q — the distance n A, 

I = length of the arbor A B, 

9 — the angle which the direction of F makes with the vertical 

or direction of the resistance Q. 
Then the force applied in the plane perpendicular to the trunnion 
A, and acting parallel to the resistance Q, will, § 95, be, 

and the force applied in this plane and acting at right angles to the 
direction of Q, will be 

F j-^ • sm 9. 

The vertical force applied in the plane at B will be . 
^•f+ Q-J + P-f •COS9, 



APPLICATIOlN^S. 391 

and the horizontal force in this plane will be 

whence, 

JSr=j' V Mi-9)+ Q{l-q)'^P{l-p)oos:pV-\-F%l-py-Sin^;p ; • (568) , 

JN''=y^[w.g + Q.q -{- F.p, cos ^y + P^ . jp^ . shi' :p -, • .(569) 

If & and &' be the angles which the directions of iV and iV^' make 
with that of the resistance Q, we have 

sin & = ^ ^ • sin 9 ; sm &' ■= -^ . sin 9. 

Equations (567), (568), and (569) are sufficient to determine the rela- 
tion between P and Q to preserve the motion uniform, or an equili- 
brium without the aid of inertia. The values of N and N' being 
substituted in Equation (567), and that equation solved with refer- 
ence to P, will give the relation in question. 

§ 333. — If the power P act in the direction of the resistance Q^ 
then will cos 9 = 1, sin 9 = 0, and Equation (567) would, after 
substituting the corresponding values of N and N\ transposing, 
omitting the common factor d s^ , and supposing p = p', become 

PE= Qr+fp{w+ Q + P) + d^. ^\^^ .r.. . (570) 

And omitting the terms involving the friction and stiffness of 
cordage, 

Q ~ P' 

that is, the power is to the resistance as the radius of the arbor 
is to that of the wheel ; . which relation is exactly the same as 
that of the common lever. 

FIXED PULLEY. 

§ 334. — The pulley is a small wheel having a groove in " its cir- 
cumference for the reception of a rope, to one end of which the 



392 



ELEMENTS OF ANALYTICAL MECHANICS. 



power P is applied, and to the other the resistance Q. The pulley 
may turn either upon trunnions or about an axle, supported in what 





is called a hlocJc. This is usually a solid piece of wood, through 
which is cut an opening large enough to receive the pulley, and 
allow it to turn freely between its cheeks. Sometimes the block is 
a simple framework of metal. When the block is stationary, the 
pulley is said to be fixed. The principle of this machine is obvi- 
ously the same as that of the wheel and axle. 

The friction between the rope and pulley will be sufficient to 
give the latter motion. 

Making, in Equations (568) and (569), 



ii, 



have 



N = ^ -y/iw + ^ -f- P cos 9)2 + P2 sin2 ^ = JSf' - - (571) 



Making E = r, and p =: p', in Equation (567), and substituting 
the above values of iV and N\ we have, after omitting the common 
factor rfsi. 



FE-QE-fp^(w-{-Q-i-Pcos(py+FHm^(p-d^'^^^^-B=0. .(572) 



APPLICATIONS. 



393 



Solving this equation with respect to P, we find the value of 
the latter in terms of the different sources of resistance. But this 
direct process would be tedious ; and it will be sufficient in all 
cases of practice to employ an approximate value for P under the 
radical, obtained by first neglecting the terms involving fiiction and 
stiffness of cordage. 

Thus, dividing by R and transposing, we find 

P = Q-\-f-^ V{^^ + Q+Pcos^f + P^sm^9 + ^r '^^j^^ ' 

Now f ' ~ is usually a small fraction ; an erroneous value as- 

sumed for P under the radical, will involve but a triffing error in 
the result. We may therefore write Q for P in the second mem- 
ber ; and neglecting the weight of the pulley, which is always in- 
significant in comparison to Q^ we have 



P = Q\\ 4-/-^^2(l +cos9)]+c/, 



2P 



but 



whence. 



1 -f cos 9 = 2 cos2 1 9 ; 



P=Q{\+ 2f'^.cosi o) + d,. ^-^ 



(573) 



(574) 



In which 9 denotes the angle A M i>, which 
is the supplement of the angle A C i>, and de- 
noting this latter angle by ^, we have 



cos J 9 = sin J ^ , 
whence 

P=(2(l+2/-^sinl^) + cZ,^-y^ 



(575) 




If the arc of the pulley, enveloped by the rope, be 180°, then 
will 



P=Q{\ -1-2/'.-^)+ c/, 



2R 



(57G) 



394: 



ELEMENTS OF ANALYTICAL MECHANICS. 



If the friction and stiffness of cordage be so small as to justify their 
omission, then will 

p=q. 

That is, the power must be equal to the resistance, and the only- 
office of the cord or rope is to change the direction of the power. 



MOVABLE PULLEY. 

§335. — In the fixed pulley, the resultant action of the power and 
resistance is thrown upon the trunnion boxes. If one end of the 
rope be attached to a fixed hook A^ 
while the power P is applied to the 
other, and the pulley is left free to roll 
along the rope, the resistance W to be 
overcome may be connected with its 
trunnion, after the manner of the figure ; 
the pulley is then said to be movable, 
and the relation between the power and 
resistance is still given by Eq. (567,) 
in which the principal resistance be- 
comes N + N\ and the tension of the 
rope between the fixed point A, and the 
tangential point H, becomes Q. 

Making in Equation (567), i2 = r, p = p', and W'= N+N'zzz^IT, 
we have 




PR- QB-f'p W-d^ 
dividing by i?, and transposing 



2B 



K+I Q 
2R 



P = 



(577) 



(578) 



Eliminating Q by means of Equation (571), and solving the resulting 
equation with respect to P, the value of the power will be known 
in terms of the resistances. The process may be much abridged by 
limiting the solution to an approximation, which will be found suffi- 
cient in practice. 



APPLICATION'S. 



395 



.Neglecting the weight of the pulley, which is always insignificant 
in comparison with P or Q^ and making Q — F, which would be the 
case if -we neglect friction and stiffness of cordage, Equation (571), 
gives 



and because 



or. 



i\r=iTF=i$ V^(l + C0S9); 

1 + cos 9 = 2 cos2 ^(p = 2 sin2 i ^, 
W =2 Q .sin id; 



W_^ 

^ 9 cin 1 A ' 



2 sin i 



which, in Equation (578), gives 



^=^(3dH-p+^'-B+"' 2 



w 



R 



-. (579) 



The quantity of work is found by multiplying both members by 
i2 Sj , in which ^i is the arc described at the unit's distance. 

If the arc enveloped by the rope be 180°, then will ^d — 90°, 
sin J 4 = 1, and 



^-^0+/'i) + 



d. 



K + jl. W 
2E 



(580) 



If the friction and stiffness of cordage be neglected, then will. 
Equation (579), 

TF =: 2 P sin .H, 

and multiplying by R^ 

i2 PT =r P . 2 i2 . sin i ^ ; 



but 

vrhence, 



2 P sin I ^ = y1 P ; 



R . W = P . AB; 

that is, the power is to the resistance as the 
radius of the pulley is to the cord of the arc 
enveloped hy the rope. 




396 



ELEMENTS OF ANALYTICAL MECHANICS 



'( 




§ 336. — The Muffie is a collection of pulleys in two separate 
blocks or frames. One of these blocks is attached to a fixed point 
A^ by which all of its pulleys become Jlxed^ 
while the other block is attached to the resist- 
ance TF", and its pulleys thereby made mov- 
able. A rope is attached at one end to a hook 
h at the extremity of the fixed block, and is 
passed around one of the movable pulleys, 
then about one of the fixed pulleys, and so on, 
in order, till the rope is made to act upon each 
pulley of the combination. The power P is 
applied to the other end of the rope, and the 
pulleys are so proportioned that the parts of 
the rope between tSem, when stretched, are 
parallel. Now, suppose the power P to main- 
tain in uniform motion the point of applica- 
tion of the resistance W; denote the tension 
of the rope between the hook of the fixed 
block and the point where it comes in con- 
tact with the first movable pulley by t-^ ; the 
radius of this pulley by R^ ; that of its eye 
by rj; the co-efRcient of friction on the axle 
by /; the constant and co-efficient of the stiff- 
ness of cordage by K and /, as before ; then, denoting the tension of 
the rope between the last point of contact with the first movable, 
and first point of contact with the first fixed pulley, by t^^ the quan- 
tity of work of the tension t^ will. Equation (515), be 




in which 



t, B, s, + d, -±3^ iJ. h + /' {k + t,) n s, ; 



2ii, 



/' 



/ 



VTT7 



dividing by 



t,B, = t,B, + d,.^±^.B,+f(,h + k)n. . (581) 



APPLICATIONS. 397 

Again, denoting the tension of that part of the rope which passes 
from the first fixed to the second movable pulley by ^'3 , the radius 
of the first fixed pulley by R^ , and that of its eye by r^ , we shall, 
in like manner, have 

t,R, = t, R, + d, ^^— ^2 + f {t, + t,) r,. . (582) 

And denoting the tensions, in order, by t^ and t., , this last being 
equal to P, we shall have 

t, R, = 4 i?3 + d, ^^^ • i?3 4- /' (^3 + Q 7-3. . (583) 

PR, = t,R, + d, ^^p B, + r {h + P) U. . (584) 

so that we finally arrive at the power P, through the tensions which 
are as yet unknown. The parts of the rope being parallel, and the 
resistance W being supported by their tensions, the latter may ob- 
viously be regarded as equal in intensity to the components of W) 
hence, 

t, + t, + t,-\-t,^W', . . . . . (585) 

which, with the preceding, gives us five equations for the determi- 
nation of the four tensions and power P. This would involve a 
tedious process of elimination, which may be avoided by contenting 
ourselves with an approximation which is found, in practice, to be 
sufficiently accurate. 

If the friction and stiffiiess be supposed zero, for the moment, 
Equations (581) to (584) become 

h ^i = ^1 ^1 > 
tz Ri = t<i R-2 ^ 
h ^3 = ^3 -^3 J 

PJR. = UR,; 
from which it is apparent, dividing out the radii i?i , i?^, /?, , dec, 



398 ELEMENTS OF ANALYTICAL MECHANICS. 

that ti = ti, h = iii h =z t^, F = t^j and hence. Equation (585) 

becomes 

whence, 

'•-IT' 

the denominator 4 being the whole number of pulleys, movable and 
fixed. Had there been n pulleys, then would 

W 
n 

With this approximate value of t^, we resort to Equations (581) 
to (584), and find the values of t^, t^, t^, &c. Adding all these 
tensions together, we shall find their sum to be greater than PF, 
and hence we infer each of them to be too large. If we now 
suppose the true tensions to be proportional to those just found, 
and whose sum is TFj > TF", we may find the true tension corre- 
sponding to any erroneous tension, as t^ , by the following propor- 
tion, viz. : 

W 

or, which is the same thing, multiply each of the tensions found by 

W 

the constant ratio r— ? the product will be the true tensions, very 

nearly. The value of ^4 thus found, substituted in Equation (584), 
will give that of P. 

Example. — Let the radii i^i , i?2 , Rz and R^ , be respectively 
0,26, 0,39, 0,52, 0,65 feet ; the radii r^ = 7\ — r^ = r^ of the 
eyes =r 0,06 feet ; the diameter of the rope, which is white and 
dry, 0,79 inches, of which the constant and co-efficient of rigidity 
are, respectively, K = 1,6097 and / — 0,0319501 ; and suppose the 
pulley of brass, and its axle of wrought iron, of which the co-efficient 
/ = 0,09, and the resistance W a weight of 2400 pounds. 

Without friction and stiffness of cordage, 

2400 ^*^- 

t, = ^ = 600. 



APPLICATIONS. 



399 



Dividing Equation (581) by B^, it becomes, since d^ z= 1, 

Substituting the value of M^ , and the above value of ti , and regard- 
ing in the last term t^ as equal to t^ , which we may do, because 

of the small co-efficient -^ /', we find 



ti = < 



600 

1,6097 + 0,0319501 x 600 

2 X (0,26) 



+ ^ X 0,09 X (600 + 600) 



+ 



;. = 628,39. 



Again, dividing Equation (582) by E,^, and substituting this value 
of ^2 a^^d that of -Rj) we find 

lbs. 

t, = 673,59. 

Dividing Equation (583) by H^ , and substituting this value of 4 , as 
well as that of B^ , there will result 



whence. 



h 



Wi = t, + t,-h tz + U 



lbs. 

709,82 



and 






2400 



600 ^ 

-f 628,39 
+ 673,59 
+ 709,82 

= 0,919; 



2611,80; 



2611,80 

which will give for the true values of 

^ = 0,919 X 600 = 551,400 
/., = 0,919 X 628,39 =.: 577,490 
^3 = 0,919 X 673,59 = 619,029 
^4 = 0,919 X 709,82 = 652,324 



2400,243 



400 



ELEMENTS OF ANALYTICAL MECHANICS. 



The above value for 4 = 652,324, in Equation (584), will give, after 
dividing by R^^ and substituting its numerical value, 



652,324 

1,6097 4- 0,03195 x 652,324 



P = < 



2 X 0,65 



0,06 



+ ^ X 0,09 X (652,324 + P) ; 



and making in the last factor P = i^ = 652,324, we find 

lbs. lbs. lbs. lbs. 

P = 652,324 + 17,270 + 10,831 = 680,425. 

Thus, without friction or stiffness of cordage, the intensity of P would 
be 600 lbs. ; with both of these causes of resistance, which cannot be 
avoided in practice, it becomes 680,425 lbs., making a difference of 
80,425 lbs., or nearly one-seventh ; and as the quantity of work of 
the power is proportional to its intensity, we see that to overcome 
friction and stiffness of rope, in the example before us, the motor 
must expend nearly a seventh more work than if these sources of 
resistance did not exist. 



THE WEDGE. 

§ 337. — The wedge is usually employed in the operation of cut- 
ting, splitting, or separating. It consists 
of an acute right triangular prism ABC. 
The acute dihedral angle A Cb is called 
the edge ; the opposite plane face A b 
the hack; and the planes Ac and Cb^ 
which terminate in the edge, the faces. 
The more common application of the 
wedge consists in driving it, by a blow 
upon its back, into any substance which 
we wish to split or divide into parts, in 
such manner that afler each advance it 
shall be supported against the faces of 
the opening till the work is accomplished. 




APPLIC ATIOXS. 



401 



§ 338. — The blow by which the wedge is driven forward will be 
supposed perpendicular to its back, for if it were oblique, it would 
only tend to impart a rotary motion, and give rise to complications 
which it would be unprofitable to consider : and to make the case 
conform still further to practice, we will suppose the wedge to be 
isosceles. 

The wedge ACB being inserted in the opening a A 5, and in con- 
tact with its jaws at a and b, we know 
that the resistance of the latter will 
be perpendicular to the faces of the 
wedge. Through the points a and b 
draw the lines aq and bp normal to 



the faces A C and B C ; from their 



point of intersection lay off the 
distances Oq and Op equal, respec- 
tively, to the resistances at a and b. 
Denote the first by Q^ and the second 
by P. Completing the parallelogram 
Oqmp^ Om will represent the re- 
sultant of the resistances Q and P. 
Denote this resultant by R\ and the 
angle ^ C^ of the wedge by ^, which, 
in the quadrilateral a Ob C^ will be 

equal to the supplement of the angle a b zzz p q, the angle made 
by the directions of Q and P. From the parallelogram of forces, 
we have. 




R' 



P'^+ Q'^^+^P Qdosp Oq - P'^-\- Q'^ -2P Q COS&; 



or. 



R' = y/p"^ + Q^ -2 P Q cos 



The resistance Q will produce a friction on the face ^4 C equal 
to fQ, and the resistance P will produce on the face B C the fric- 
tion / P : these act in the directions of the faces of the wedge. 
Produce them till they meet in C, and lay off the distances C q' and 
Cp' to represent their intensities, and complete the parallelogram 

26 



402 ELEMENTS OF ANALYTICAL MECHANICS. 

Cq' 0' p' \ CO' will represent the resultant of the frictions. Denote 
this by jR"j and we have, from the parallelogram of forces, 

JS"2 ^p Q2 J^ p p2 + 2/2 P ^ COS ^ ; 

or, 

R" ^f -y/F'' + $2 + 2P ^cosl 

The wedge being isosceles, the resistances P and Q will be equal, 
their directions being normal to the faces will intersect on the line 
Ci>, which bisects the angle (7 = ^, and their resultant will coin- 
cide with this line. In like manner the frictions will be equal, and 
their resultant will coincide with the same line. Making Q and P 
equal, we have, from the above equations. 



R' = P '/2 (1 - cos ^), 



R" = fP -v/2 (1 + cos^). 

But, 

1 — cos ^ = 2 sin2 1 d, 

1 + cos ^ = 2 cos2 1. ^ J 
whence we obtain, by substituting and reducing, 
R' = 2P. sin i &, 
P"= 2/.P. cosi^; 



and further, 



therefore, 



cos J ^ = ZC" ' 



R' = P 



R"= 2f'P 



A C" 
CD 



A C 



Denote by F the intensity of the blow on the back of the wedge. 
If this blow be just sufficient to produce an equilibrium bordering 



APPLICATIOXS. 403 

on motion forward, call it F' ; the friction will oppose it, and we 
must have, 

ir' = i2'+i2"=:P.^+2/.P.-^. . . . (586) 

If. on the contrarr, the blow be just sufficient to pre^^ent the wedge 
from flying back, call it F" ; the friction will aid it, and we must 
have, 

^"=^'4^-^^-^-S • • • • (^^') 

The wedge will not move under the action of any force whose inten- 
sity is between F' and F''. Any force less than F"^ will allow it 
to fly back; any force greater than F\ will drive it forward. Tlie 
range through which the force may vary without producing motion, 
is obviously, 

F'-F"^ifP-^ (588) 

which becomes greater and greater, in proportion &?, C D and A C 
become more nearly equal ; that is to say, in proportion as the 
wedges becomes more and more acute. 

The ordinary mode of employing the wedge requires that it shall 
retain of itself whatever position it may be driven to. This makes 
it necessary that F" should be zero or negative, Eq. (587), M'hence 



P-44r=^U-P-^,o..P^<.f.P -^ 



AC ' AC A C ^ ' AC 



or, omitting the common factors and dividing both members of the 
equation and inequality by ^ C D^ 

^A B ^ lAB 

- A B 
but ^—p-R ^^ ^^^ tangent of the angle A CD; hence we conclude, 

that the wedge will retain its place when its semi-angle does not 
exceed that whose tangent is the co-efficient of friction between the 
surface of the wedge and the surface of the opening which it is 
intended to enlarge. 



404: 



ELEMENTS OF ANALYTICAL MECHANICS. 



Resuming Eq. (587), and supposing the last term of the second 
member greater than the first term, F" becomes negative, and will 
represent the intensity of the force necessary to withdraw the wedge ; 
which will obviously be the greatest possible when ^ ^ is the least 
possible. This explains why it is that nails retain with such perti- 
nacity their places when driven* into wood, &c. 



THE SCEEW. 



§ 339. — The Screw^ regarded as a mechanical power, is a device by 
which the principles of the inclined plane are so applied as to pro- 
duce considerable pressures with great steadiness and regularity of 
motion. 

To form an idea of the figure of a screw and its mode of action, 
conceive a right cylinder, a k, with circular base, and a rectangle, of 
other plane figure, abcm, having one of its sides 
ab coincident with a surface element, while its 
plane passes through the axis of this cylinder. 
Next, suppose the plane of the generatrix to 
rotate uniformly about the axis, and the gener- 
atrix itself to move also uniformly in the direc- 
tion of that line ; and let this twofold motion 
of rotation and of translation be so regulated, 
that in one entire revolution of the plane, the 
generatrix shall progress in the direction of 

the axis over a distance greater than the side ab, which is in the 
surface of the cylinder. The generatrix will thus generate a pro- 
jecting and winding solid called a JUIet, leaving between its turns 
a groove called the channel. Each point as m in the perimeter 
of the generatrix, will generate a curve called a helix, and it is 
obvious, from what has been said, that every helix will enjoy this 
property, viz. : any one of its points as w, being taken as an origin 
of reference, as well for the curve itself as for its projection on a 
plane through this point and at right angles to the axis, the distances 
d' m\d" m'\ &;c., of the several points of the helix from this plane, 




APPLICATIONS. 



405 



are respectively proportioned to the circular arcs md\ md", <kc., 
into which the portions mm', mm", &c., of the helix, between the 
origin and these points, are projected. 

. The solid cylinder about which the fillet is wound, is called 
the newel of the screw; the distance mm"\ between the consecu- 
tive turns of the same helix, estimated in the direction of the axis, 
is called the helical interval. 

The fillet is often generated by the motion of a triangle with 
one of its sides coincident with a h ; and as the discussion will be 
more general by considering this mode of generation, we shall adopt 
it. The surfaces of the fillet, which are generated by the inclined 
faces of the triangle, are each made up of an infinite number of 
helices, all of which have the same interval, though the helices 
themselves are at different distances from the axis, and have different 
inclinations to that line. 

The inclination of the different helices to the axis of the screw, 
increases from the newel to the exterior surface of the fillet, 
the same helix preserving its 
inclination unchanged throughout. 
The scrp.w is received into a hole 
in a solid piece B of metal or 
wood, called a nut or hurr. The 
surface of the hole through the 
nut is furnished with a winding 
fillet of the same shape and size 
as the channel of the screw, so 
that the surfaces of the screw and 
nut are brgught into accurate con- 
tact. 

From this arrangement it is 
obvious that when the nut is sta- 
tionary, and a rotary* motion iS 
communicated to the screw, the 
latter will move in the direction 

of its axis ; also, when the screw is stationary and the nut is 
turned, the nut must also move in the direction of the axis. In 




406 



ELEMENTS OF ANALYTICAL MECHANICS. 



the first case, one entire revolution of the screw will carry it lon- 
gitudinally through a distance equal to the helical interval, and any 
fractional portion of an entire revolution will carry it through a pro- 
portional distance ; the same of the nut, when the latter is mova- 
ble and the screw stationary. The resistance Q is applied either to 
the head of the screw, or to the nut, depending upon which is the 
movable element ; in either case it acts in the direction D C of 
•the axis. The power F is applied at the extremity of a bar GH 
connected with the screw or nut, and acts in a plane at right 
angles to the axis of the screw. 

From the description of the screw and its mode of generation, 
"we may find the equation of its fillet or helicoidal surface. For 
this purpose, take the axis z to coincide with the axis of the newel, 
and the initial position of the generatrix in the plane t/z. Make 
s = any definite portion C C 

of an assumed helLx ; 
(p = the angle YAi, through 
which the rotating plane 
has turned during the gene- 
ration of s ; 
r = the distance CD of this 

helix from the axis z ; 
a = the angle which this helix 
makes with the plane x y ; 
§ = the angle CBD which the 
generatrix of the helicoidal 
surface makes with the 
axis z ; 
y = the co-ordinate ^^ of the 
point in which the genera- 
trix, in its initial position, intersects the axis z. 
Then, for any point as C of the generatnx in its initial position, 
we have 

z = AD=AB-\-£D = y-\-r. cotan g, 
and for any subsequent position, as C B', 

z = y -^ r . cotan to + r . 9 . tan a, . • . . (589) 




APPLICATIONS. 4:07 

•which is the equation sought, and in which a and r are constant 
for the same helix, and variable from one helix to another. 

The power P acts in a direction perpendicular to the axis of 
the newel. Denote by I its lever arm ; its virtual moment will be 

The resistance Q acts in the direction of the axis of the newel ; 
its virtual moment will be 

Qdz, 

The friction acts in the direction of the helicoidal surface and paral- 
lel to the helices. Conceive it to be concentrated upon a mean 
helix, of which the distance from the newel axis is r, and length s : 
denote the normal pressure by N^ and co-efficient of friction by /. 
The virtual moment of friction will be 

f-N.ds', 
and Equation (508), 

Pldc^ — Qdz —f. N .ds = (590) 

But the displacement must satisfy Equation (589), or, as in § 213, 
the condition, 

Z = — r . 9 . tan ol — r . cotan § — y = ; . (591) 

and also, 

r = constant (592) 

Differentiating, w^e have, 

dz — cotan ^ . d r — r tan a c?(p = 0, 
dr = 0. 

Multiplying the first by X, the second by X', adding to Equation 
(590), and eliminating d s hy the relation 

d s — r . d (p . cos a -{- d z . sin oi, . . . . (593) 
we find, 
{P I — /. iV. cos a .r - X tana.r)(/f + (X - - /.i\^. sin a)dz + (\'- X cotaa€)(/r = 



408 ELEMENTS OF AXALYTICAL MECHAXICS. 
and, from the principle of indeterminate co-efficients, 

P Z — / . -lY . cos a . ?• — X . tan a . r = ; . . (594) 

Q +f^\smoi -\ = 0; (595) 

X' — X cotan b = (595)' 

The variables d z, d r, emdrdcp, are rectangular; whence. Equation 
(331), 

Substituting this in Equations (594) and (595). and eliminating X, 
there will result 



^ ^ r tan a + /". cos a . -1/1+ tan^ a -{- cotan'^ b ,^^^. 

F = Q • J ' ■ (o96) 

^ 1 — / . sin a . yi -r tan^ a + cotan'- « 

Substituting the value of X from Equation (595), in Equation 
(595)', we find, 

r=Q ;^ ' .; . (597) 

1 — / . sin a y 1 -f tan^ a -f cotan^ o 

in which X' is, § 217, the value of the force acting in the direction 
of r. 

§ 340.— If the fillet be rectangular, § — 90^ cotan § = 0, and 



P= ^.!:.tana+/.cosaVl + tan3a^ ^ ^^^^^ 
^ 1 — / . sin a . -y/l 4- tan^ a 

and 

X' r= 0. 

§341. — If we neglect the friction, /= 0; and 
Fl = Q . r . tan a, 
multiplying both members by 2 -^r, 

P.2cr/:= g.2crr. tan a (599) 

That is, the power is to the resistance as the helical interval is to 
the circumference described by the end of the lever arm of the power. 



APPLICATIONS. 



409 



PUMPS. 



§ 342. — Any machine used for raising liquids from one level 
to a higher, in which the agency of atmospheric pressure is employed, 
is called a Pump. There are various kinds of pumps ; the more 
common are the sucking^ forcing^ and lifting pumps. 

§ 343. — The Sucking -Pump consists of a cylindrical body or barrel 
j5, from the lower end of which a tube D^ called the sucking-pipe, 
descends into the water contained in a reservoir or well. In the 
interior of the barrel is a movable piston C, surrounded with leather 
to make it water-tight, yet ca- 
pable of moving up and down 
freely. The piston is perforated 
in the direction of the bore of 
the barrel, and the orifice is 
covered by a valve F called 
the piston-valve^ which opens up- 
ward ; a similar valve E^ called 
the sleeping-valve^ at the bottom 
of the barrel, covers the upper 
end of the sucking-pipe. Above 
the highest point ever occupied 
by the piston, a discharge-pipe 
P is inserted into the barrel ; 
the piston is worked by means 
of a lever //, or other contriv- 
ance, attached to the piston-rod 

G. The distance yl J', between the highest and lowest points of the 
piston, is called the play. To explain the action of this pump, let 
the piston be at its lowest point A^ the valves E and F closed by 
their own weight, and the air within the pump of the same density 
and elastic force as that on the exterior. The water of the reservoir 
will stand at the same level L L both within and without the 
sucking-pipe. Now suppose the piston raised to its highest ])oiiit A\ 
the air contained in the barrel and sucking-pipe will tend by its 




4:10 ELEMENTS OF ANALYTICAL MECHANICS. 

elastic force to occupy the space which the piston leaves void, the 
valve E will, therefore, be forced open, and air will pass from the 
pipe to the barrel, its elasticity diminishing in proportion as it fills 
a larger space. It will, therefore, exert a less pressure on the 
water below it in the sucli:ing-pipe than the exterior air does on that 
in the reservoir, and the excess of pressure on the part of the 
exterior air, will force the water up the pipe till the weight of the 
suspended column, increased by the elastic force of the internal air, 
becomes equal to the pressure of the exterior air. When this takes 
place, the valve U will close of its own weight ; and if the piston 
be depressed, the air contained between it and this valve, having 
its density augmented as the piston is lowered, will at length have 
its elasticity greater than that of the exterior air ; this excess of 
elasticity will force open the valve F^ and air enough will escape 
to reduce what is left to the same density as that of the exterior 
air. The valve F will then fall of its own weight ; and if the 
piston be again elevated, the water will rise still higher, for the 
same reason as before. This operation of raising and depressing 
the piston being repeated a few times, the w^ater will at length enter 
the barrel, through the valve F^ and be delivered from the dis- 
charge-pipe P. The valves E and F^ closing after the water has 
passed them, the latter is prevented from returning, and a cylinder 
of water equal to that through which the piston is raised, will, at 
each upward motion, be forced out, provided the discharge-pipe is 
large enough. As the ascent of the water to the piston is pro- 
duced by the difference of pressure of the internal and external air, 
it is plain that the lowest point to which the piston may reach, 
should never have a greater altitude above the water in *the reser^ 
voir than that of the column of this fluid which the atmospheric 
pressure may support, in vacuo, at the place. 

§ 344. — It will readily appear that the rise of water, during 
each ascent of the piston after the first, depends upon the expulsion 
of air through the piston-valve in its previous descent. But air can 
only issue through this valve when the air below it has a greater 
density and therefore greater elasticity than the external air ; and 



APPLICATIONS. 



411 



if the piston may not descend low enough, for want of sufficient 
play, to produce this degree of compression, the water must cease 
to rise, and the working of the piston can have no other efitct than 
alternately to compress and dilate the same 
air between it and the surface of the water. 
To ascertain, therefore, the relation which the 
play of the piston should bear to the other 
dimensions, in order to make the pump effec- 
tive, suppose the water to have reached a sta- 
tionary level X, at some one ascent of the 
piston to its highest point A', and that, in its 
subsequent descent, the piston-valve will not 
open, but the air below it will be compressed 
only to the same density with the external air 
when the piston reaches its lowest point A. 
The piston may be worked up and down in- 
definitely, within these limits for the play, 
without moving the water. Denote the play 

of the piston by a ; the greatest height to which the piston may be 
raised above the level of the water in the reservoir, by b. which niay 
also be regarded as the altitude of the discharge pipe ; the elevation 
of the point X, at which the water stops, above the water in the 
reservoir, by x ; the cross-section of the interior of the barrel by B. 
The volume of the air between the level X and A will be 




B X (b - X - a)', 

the volume of this same air, when the piston is raised to A', pro- 
vided the water does not move, will be 

B (b - x). 

Represent by h the greatest height to which water may be supported 
in vacuo at the place. The weight of the column of water wliich 
the elastic force of the air, when occupying the space between the 
limits X and A, will support in a tube, with a bore equal to that 
of the barrel is measured by 



Bh.ff.B; 



412 ELEMENTS OF ANALYTICAL MECHANICS. 

in which D is the density of the water, and g the force of gravity. 
The weight of the column which the elastic force of this same air 
will support, when expanded between the limits X and A\ will be 

Bh'.g.D', 

in which h' denotes the height of this new column. But, frv.m Ma- 
riotte's law, we have 

B{b — X — a) : B{b — x) : -. B h' g D : Bhg D; 
whence, 

— X 

But there is an equilibrium between th.e pressure of the external 
air and that of the rarefied air between the limits X and A\ when 
the latter is increased by the weight of the column of water whose 
altitude is x. Whence, omitting the common factors B^ D and g^ 

X ■-{- h' = X -\- h —^ = k ; 

— X 

or, clearing the fraction and solving the equation in reference to x, 
we find 



X z=: ib dt ^ ,/b^ - 4ah. ..... (600) 

When X has a real value, the water will cease to rise, but x 
will be real as long as 6^ is greater than 4 ah. If, on the con- 
trary, 4 ah is greater than b^, the value of x will be imaginary, and. 
the water cannot cease to rise, and the pump will always be effective 
when its dimensions satisfy this condition, viz. : — 

4a A > 62, 



Th 



« > . . , 



that is to say, the play of the piston must be greater than the square 
of the altitude of the upper limit of the play of the 2;z5/o?z above 
the surface of the water in the reservoir, divided by four times the 
height to which the atmospheric pressure at the place, where the pump 



APPLICATIONS. 



413 



is used, will support water in vacuo. This last height is easily found 
by means of the barometer. We have but to notice the altitude 
of the barometer at the place, and multiply its column, reduced to 
feet, by 13|-, this being the specific gravity of mercury referred to 
water as a standard, and the product will give the value of h in 
feet. 

Example. — Required the least play of the piston in a sucking- 
pump intended to raise water through a height of 13 feet, at a 
place where 'the barometer stands at 28 inches. 



Here 

Barometer, 

Play 



6 := 13, and 62 

in. 

28 



169. 



12 



= 2,333 feet. 



ft- 



h = 2,333 X 13,5 = 31,5 feet. 



62 



169 



^ -^ 4A 4 X 31.5 



ft. 
= 1,341 + ; 



that is, the play of the piston must be greater than one and one 
third of a foot. -j-iiroJ ' . 

§ 345. — The quantity of work performed by 
the motor during the delivery of water through 
the discharge-pipe, is easily computed. Sup- 
pose the piston to have any position, as M, 
and to be moving upward, the water being 
at the level LL in the reservoir, and at P 
in the pump. The pressure upon the upper 
surface of the piston will be equal to the 
entire atmospheric pressure denoted by A, 
increased by the weight of the column of 
water MF\ whose height is c', and whose 
base is the area B of the piston ; that is, the 
pressure upon the top of the piston will be 

A + Bc'ffD, 

in which r/ and D are the force of gravity and density of the water, 
respectively. Again, the pressure upon the under surface of the 



y 




JP 




1 


■ 


M. 


xir 


F^^ 


o 



414: ELEMENTS OF ANALYTICAL MECHANICS. 

piston is equal to the atmospheric pressure A, transmitted through 
the water in the reservoir and up the suspended column, diminished 
by the weight of the column of water JVM below the piston, and 
of which the base is £ and altitude c ; that is, the pressure fronx 
below will be 

A — BcgD, 

and the difference of these pressures will be 

A -f Be' g D — {A - Beg D) = B g D {c + e') ; 

but, employing the notation of the sucking-pump just described, 

c + e' = b', 

whence, the foregoing expression becomes 

Bb.g.B; 

which is obviously the weight of a column of the fluid whose base 
is the area of the piston and altitude the height of the discharge-pipe 
above the level of the water in the reservoir. And adding to this 
the effort necessary to overcome the friction of the parts of the pump 
when in motion, denoted by 9, we shall have the resistance which the 
force i^, applied to the piston-rod, must overcome to produce any- 
useful effect; that is, 

F = BbgD + 9. 

Denote the play of the piston by p^ and the number of its double 
strokes, from the beginning of the flow through the discharge-pipe 
till any quantity Q is delivered, by n ; the quantity of work will, by 
omitting the effort necessary to depress the piston, be 

Fnp = np [B b . g I) -\- (p]', 

or estimating the volume in cubic feet, in which case p and b must 
be expressed in linear feet and B in square feet, and substituting for 
g D its value 62,5 pounds, we finally have for the quantity of work 
necessary to deliver a number of cubic feet of water Q = B np, 

Fnp = np [62,6 . Bb -i- cp]; .... (601) 

in which 9 must be expressed in pounds, and may be determined 



APPLICATION'S. 415 

either by experiment in each particular pump, or computed by the 
rules already given. 

It is apparent that the action of the sucking-pump must be very 
irregular, and that it is only during the ascent of the piston that it 
produces any useful effect ; during the descent of the piston, the force 
is scarcely exerted at all, not more than is necessary to overcome 
the friction. 

§ 346. — The Lifting-Pump does not differ much from the sucking- 
pump just described, except that the barrel and sleeping-valve E are 
placed at the bottom of the pipe, and some distance below the sur- 
face of the water Z Z in the reservoir ; the 
piston may or may not be below this 
same surface when at the lowest point of 
its play. The piston and sleeping-valves 
open upward. Supposing the piston at its 
lowest point, it will, when raised, lift the 
column of water above it, and the pres- 
sure of the external air, together with the 
head of fluid in the reservoir above the 
level of the sleeping-valve, will force the 
latter open ; the water will flow into the 
barrel and follow the piston. When the W^,;/ .^ 

piston reaches the upper limit of its play, ^i ' F 

the sleeping-valve will close and prevent ^^ 

the return of the water above it. The 

piston being depressed, its valves F will open and the water will 
flow through them till the piston reaches its lowest point. The 
same operation being repeated a few times, a column of water will 
be lifted to the mouth of the discharge-pipe P, after which every 
elevation of the piston will deliver a volume of the fluid equal to 
that of a cylinder whose base is the area of the piston and whose 
altitude is equal to its play. 

As the water on the same level within and without the pump 
will be in equilibrio, it is plain that the resistance to be overcome 
by the power will be the friction of the rubbing surflioes of the pump, 




416 



ELEMENTS OF ANALYTICAL MECHANICS. 



augmented by the weight of a column of fluid whose base is the area 
of the piston, and altitude the difference of level between the surface 
of the water in the reservoir and the discharge-pipe. Hence the 
quantity of work is estimated by the same rule, Equation (601). If 
we omit for a moment the consideration of friction, and take but a 
single elevation of the piston after the water has reached the dis- 
charge-pipe, n will equal one, 9 will be zero, and that equation re- 
duces to 

F:P = 62,5 Bp X b', 

but 62,5 X Bp is the quantity of fluid discharged at each double 
stroke of the piston, and 5 being the elevation of the discharge-pipe 
above the water in the reservoir, we see that the work will be the 
same as though that amount of fluid had actually been lifted through 
this vertical height, which, indeed, is the useful effect of the pump 
for every double stroke. 

§ 347.— The Forcing-Pump 
is a further modification of 
the simple sucking-pump. The 
barrel B and sleeping-valve 
F are placed upon the top 
of the sucking-pipe M. The 
piston F is without per- 
foration and valve, and the 
water, after being forced into 
the barrel by the atmospheric 
pressure without, as in the suck- 
ing-pump, is driven by the de- 
pression of the piston through 
a lateral pipe H into an air- 
vessel iV, at the bottom of 
which is a second sleeping-valve 
E\ opening, like the first, up- 
ward. Through the top of the 
air-vessel a discharge-pipe K 
passes, air-tight, nearly to the 




APPLICATIONS. 417 

bottom. The water, when forced into the air-vessel by the de- 
scent of the piston, rises above the lower end of this pipe, 
confines and compresses the air, which, reacting by its elas- 
ticity, forces the water up the pipe, while the valve E' is closed by 
its own weight and the pressure from above, as soon as the piston 
reaches the lower limit of its play. A few strokes of the piston will, 
in general, be sufficient to raise water in the pipe K to any desired 
height, the only limit being that determined by the power at com- 
mand and the strength of the pump. 

§ 348. — During the ascent of the piston, the valve E' is closed 
and E is open ; the pressure upon the upper surface of the piston 
is that exerted by the entire atmosphere ; the pressure upon the 
lower surface is that of the entire atmosphere transmitted from the 
surface of the reservoir through the fluid up the pump, diminished 
by the weight of the column of water whose base is the area of 
the piston and altitude the height of the piston above the surface 
of the water in the reservoir ; hence, the resistance to be overcome 
by the power will be the difference of these pressures, which is 
obviously the weight of this column of water. Denote the area 
of the piston by B^ its height above the water of the reservoir at 
one instant by y, and the weight of a unit of volume of the fluid 
by w^ then will the resistance to be overcome at this point of the 
ascent be 

w.B.y; 

and the elementary quantity of work will be 

w . B .ydy, 

and the whole work during the ascent will be 

w . Bj''y dy = wB- ^-^^ (,/ -y,); 

in which y' and y^ are the distances of the upper and lower limits 
of the play of the piston from the water in the reservoir. 

But B .[y' — y^) is the volume of the barrel within the limits 
of the play of the piston, and i (y' -\- y^) is the height of its centre 
of gravity above the level of the fluid in the reservoir. 

27 



418 ELEMENTS OF ANALYTICAL MECHANICS. 

y' -\- y 
Denoting the play by ^, and making — - — - = z\ we have for 

the quantity of work during the ascent, 

w . B .p . z\ 

During the descent of the piston, the valve £J is closed, and E^ 
open, and as the columns of the fluid in the barrel and discharge- 
pipe, below the horizontal plane of the lower surface of the piston, 
will maintain each other in equilibrio, the resistance to be over- 
come by the power will be the weight of a column of fluid whose 
base is the area of the piston and altitude the difference of level 
between the piston and point of delivery F ; and denoting by z. 
the distance of the central point of the play below the point P, 
we shall find, by exactly the same process, 

wBpz,, 

for the quantity of work of the motor during the descent of the 
piston ; and hence the quantity of work during an entire double 
stroke will be the sum of these, or 

w Bp (z' + z^. 

But z' 4- z^ is the height of the point of delivery P above the 
surface of the water in the reservoir ; denoting this, as before, by 
5, we have 

w Bp b ; 

and calling the number of double strokes n, and the whole quantity 
of work Q, we finally have 

Q = nwBpb. (002) 

If we make z^ =2', or b z=z 'Hz^^ which will give z^ = -—5 the 

quantity of work during the ascent will be equal to that during 
the descent, and thus, in the forcing-pump, the work may be equalized 
and the motion made in some degree regular. In the lifting and 
sucking-pumps the motor has, during the ascent of the piston, to 
overcome the weight of the entire column whose base is equal to 
the area of the piston and altitude the diflference of level between 



APPLICATIONS. 



419 



the water in the reservoir and point of delivery, and being wholly 
relieved during the descent, when the load is thrown upon the 
sleeping-valve and its box, the work becomes variable, and the 
motion irregular. 



THE SIPHON. 




§ 349. — The Siphon is a bent tube of unequal branches, open at 
both ends, and is used to convey a liquid 
from a higher to a lower level, over an in- 
termediate point higher than either. Its 
parallel branches being in a vertical plane 
and plunged into two liquids whose upper 
surfaces are at L M and L' M\ the fluid 
will stand at the same level both within 
and without each branch of the tube when 
a vent or small opening is made at 0. 
If the air be withdrawn from the siphon 
through this vent, the water will rise in the 

branches by the atmospheric pressure without, and when the two 
columns unite and the vent is closed, the liquid will flow from the 
reservoir A to A', as long as the level L' M' is below L 1/, and the 
end of the shorter branch of the siphon is below the surface of the 
liquid in the reservoir A. 

The atmospheric pressures upon the surfaces L M and L' M\ 
tend to force the liquid up the two branches of the tube. When 
the siphon is filled with the liquid, each of these pressures is coun- 
teracted in part by the pressure of the fluid column in the branch 
of the siphon that dips into the fluid upon which the pressure is 
exerted. The atmospheric pressures are very nearly the same for a 
difleroiice of level of several feet, by reason of the slight density 
of air. The pressures of the suspended columns of water will, for the 
same diffl-rence of level, difl'cjr considerably, in consequence of the 
greater density of the liquid. The atmospheric pressure opposed 
to the weight of the longer colunm will therefore be more counter- 
acted than that opposed to the weight of the shorter, thus leaving 



420 ELEMENTS OF ANALYTICAL MECHANICS. 

an excess of pressure at the end of the shorter branch, which will 
produce the motion. Thus, denote by A the intensity of the at- 
mospheric pressure upon a surface a equal to that of a cross-section 
of the tube ; by h the difference of level between the surface L M 
and the bend 0; by A' the difference of level between the same 
point and the level L' M' -, hj D the density of the liquid; 
and by ff the force of gravity: then will the pressure, which tends 
to force the fluid up the branch which dips below L M^ be 

A — ah J) g\ 

and that which tends to force the fluid up the branch immersed 
in the other reservoir, be 

A — ah' Dg-, 
and subtracting the first from the second, we find 

aDg{h'-h\ 

for the intensity of the force which urges the fluid within the 
siphon, from the upper to the lower reservoir. 

Denote by I the length of the siphon from one level to the 
other. This will be the distance over which the above force will 
be instantly transmitted, and the quantity of its work will be 
measured by 

aDg{h' ^ h)l 

The mass moved will be the fluid in the siphon which is measured 
by alD', and if we denote the velocity by F, we shall have, for the 
living force of the moving mass, 

alD. F2; 
whence, 

aDg{h' - h)l= ; 

and, 



V= V2^(A' -A); 

from which it appears, that the velocity with which the liquid will 
flow through the siphon^ is equal to the square root of twice the force 
of gravity^ into the difference of level of the fluid in the two reser- 



APPLICATIONS. 



4:21 



voirs. When the fluid in the reservoirs comes to the same level, 
the flow will cease, since, in that case, h' — h = 0. 

§350. — The siphon may be employed to great advantage to 
drain canals, ponds, marshes, and the like. Por this purpose, it may 
be made flexible by constructing it 
of leather, well saturated with 
grease, lil^e the common hose, and 
furnished with internal hoops to 
prevent its collapsing by the pres- 
sure of the external air. It is 
thrown into the water to be drained, 
and filled ; when, the ends being 
plugged up, it is placed across the 

ridge or bank over which the water is to be conveyed ; the plugs 
are then removed, the flow will take place, and thus the atmos- 
phere will be made literally to press the water from one basin to 
another, over an intermediate ridge. 

It is obvious that the diflerence of level between the bottom of 
the basin to be drained and the highest point 0, over which the 
water is to be conveyed, should never exceed the height to which 
water may be supported in vacuo by the atmospheric pressure at 
the place. 




THE AIR-PUMP. 



§ 351. — Air expands and tends to diff'iise itself in all directions 
when the surrounding pressure is lessened. By means of this pro- 
perty, it may be rarefied and brought to almost any degree of tenu- 
ity. This is accomplished by an instrument called the Air-Fumj) or 
Exhausting Syringe. It will be best understood by describing one 
of the simplest kind. It consists, essentially, of 

1st. A Receiver H, or chamber from which the exterior air is ex- 
cluded, that the air within may be rarefied. This is commonly a 
bell-shaped glass vessel, with ground edge, over which a small quan- 
tity of grease is smeared, that no air may pass through any renuiin- 



422 



ELEMENTS OF ANALYTICAL MECHANICS. 




ing inequalities on its surface, and a ground glass plate m n imbedded 
in a metallic table, on which it stands. 

2d. A Barrel £, 
or chamber into 
which the air in 
the reservoir is to 
expand itself. It 
is a hollow cylin- 
der of metal or 
glass, connected 
with the receiver 
i? by the commu- 
nication offf. An 

air-tight piston F is made to move back and forth in the barrel by 
means of the handle a. 

3d. A Stop-cock h, by means of which the communication between 
the barrel and receiver is established or cut off at pleasure. This 
cock is a conical piece of metal fitting air-tight into an aperture 
just at the lower end of the barrel, and is pierced in two directions ; 
one of the perforations runs transversely through, as shown in the 
first figure, and when in this position the communication between 
the barrel and re- 
ceiver is estab- 
lished ; the second 
perforation passes 
in the direction of 
the axis from the 
smaller end, and ^ 

as it approaches 

the first, inclines sideways, and runs o^it at right angles to it, as 
indicated in the second figure. In this position of the cock, the 
communication between the receiver and barrel is cut off, whilst 
that with the external air is opened. 

Now, suppose the piston at the bottom of the barrel, and the 
communication between the barrel and the receiver established; 
draw the piston back, the air in the receiver will rush out in the 




APPLICATIONS. 423 

direction indicated by the arrow-head, through the communication 
o/y, into the vacant space ^vithin the barrel. The air Avhich now 
occuiDies both the barrel and receiver is less dense than when it occu- 
pied the receiver alone. Turn the cock a quarter round, the com- 
munication between the receiver and barrel is cut off, and that be- 
tween the latter and the open air is established; push the piston to 
the bottom of the barrel again, the air within the barrel vrill be 
delivered into the external air. Turn the cock a quarter back, the 
communication between the barrel and receiver is restored ; and 
the same operation as before being repeated, a certain quantity of 
air will be transferred from the receiver to the exterior space at 
each double stroke of the piston. 

To" find the degree of exhaustion after any number of double 
strokes of the piston, denote by D the density of the air in the re- 
ceiver before the operation begins, being the same as that of the 
external air ; by r the capacity of the receiver, by h that of the bar- 
rel, and by jp that of the pipe. At the beginnmg of the operation, 
the piston is at the bottom of the barrel, and the internal air occu- 
pies the receiver and pipe; when the piston is withdrawn to the 
opposite end of the barrel, this same air expands and occupies the 
receiver, pipe, and barrel ; and as the density of the same body is 
inversely proportional to the space it occupies, we shall have 

in which x denotes the density of the air after the piston is drawn 
back the first time. From this proportion, we fiijd 

X z^ V 



r -{- 2^ -{- b 

The cock being turned a quarter round, the piston pushed back to 
the bottom of the barrel, and the cock again turned to open the 
communication with the receiver, the operation is repeated upon the 
air whose density is a:, and we have 

r + p ^- b : r + p : : D- ,'' "^ ^ ^ : x' ; 

in which x' is the density after the second backward motion -of the 
piston, or after the second double stroke ; and we find 



424: 



ELEMENTS OF ANALYTICAL MECHANICS. 



.'=D.{-- 



r -\r p 






and if n denote the number of double strokes of the piston, and 
x^ the corresponding density of the remaining air, then will 

"^^ - ^ \r ^ p + bJ ' 

From which it is obvious, that although the density of the air will 
become less and less at every double stroke, yet it can never be 
reduced to nothing, however great n may be ; in other words, the 
air cannot be wholly removed from the receiver by the air-pump. 
The exhaustion will go on rapidly in proportion as the barrel is 
large as compared with the receiver and pipe, and after a few double 
strokes, the rarefaction will be sufficient for all practical purposes. 
Suppose, for example, the receiver to contain 19 units of volume, the 
pipe 1, and the barrel 10; then will 

r +p ^20^2. 

r -\- p -{■ b 30 ~ ^ * 

and suppose 4 double strokes of the piston ; then will ti = 4, and 



( r +p \^ 
\r -\- p ^ b) 



(f)^ =^ = 0,197, nearly; 



that is, after 4 double strokes, the density of the remaining air will 
be but about two tenths of the original density. With the best 
machines, the air may be rarefied from four to six hundred times. 

The degree of rarefaction is indicated in a very 
simple manner by what are called gauges. These 
not only indicate the condition of the air in the 
receiver, but also warn the operator of any leakage 
that may take place either at the edge of the receiver 
or in the joints of the instrument. The mode in 
which the gauge acts, will be readily understood from 
the di-:cus£ion of the barometer ; it will be suffi- 
cient here simply to indicate its construction. In its '^ "^ 
more perfect form, it consists of a glass tube, about 60 inches long, 
bent in the middle till the straight portions are parallel to each 
other ; one end is closed, and the branch terminating in this end is 



APPLICATIOXS. 



425 



filled with mercury. A scale of equal parts is placed between the 
branches, having its zero at a point midway from the top to the 
bottom, the numbers of the scale increasing in both directions. It 
is placed so that the branches of the tube shall be vertical, with 
its ends upward, and inclosed in an inverted glass vessel, which 
communicates with the receiver of the air-pump. 

Repeated attempts have been made to bring the air pump to 
still higher degrees of perfection since its first invention. Self-acting 
valves, opening and shutting by the elastic force of the air, have 
been used instead of cocks. Two barrels have been employed in- 
stead of one, so that an uninterrupted and more rapid rarefaction 
of the air is brought about, the piston in one barrel being made 
to ascend while that of the other descends. The most serious defect 




was that by which a portion of the air was retained between the 
piston and the bottom of the barrel. This intervening space is filled 
with air of the ordinary density at each descent of the piston ; 



426 ELEMENTS OF ANALYTICAL MECHANICS. 

when the cock is turned, and the communication re-established with 
the receiver, this air forces its way in and diminishes the rarefac- 
tion already attained. If the air in the receiver is so far rarefied, 
that one stroke of the piston will only raise such a quantity as 
equals the air contained in this space, it is plain that no further 
exhaustion can be effected by continuing to pump. This limit to 
rarefaction will be arrived at the sooner, in proportion as the 
space below the piston is larger; and one chief point in the im- 
provements has been to diminish this space as much as possible. 
A£ is a highly polished cylinder of glass, which serves as the bar- 
rel of the pump ; within it the piston works perfectly air-tight. The 
piston consists of washers of leather soaked in oil, or of cork 
covered with a leather cap, and tied together about the lower end 
C of the piston-rod by means of two parallel metal plates. The 
piston-rod Cb, which is toothed, is elevated and depressed by means 
of a cog-wheel turned by the handle M. If a thin film of oil be 
poured upon the upper surface of the piston the friction will be 
lessened, and the whole will be rendered more air-tight. To diminish 
to the utmost the space between the bottom of the barrel and the 
piston-rod, the form of a truncated cone is given to the latter, so 
that its extremity may be brought as nearly as possible into abso- 
lute contact with the cock U; this space is therefore rendered indefi- 
nitely small, the oozing of the oil down the barrel contributing still 
further to lessen it. The exchange- cock £J has the double bore 
already described, and is turned by a short lever, to which motion 
is communicated by a rod c d. The communication G H is carried 
to the two plates / and K, on one or both of which receivers may 
be placed ; the two cocks JV and below these plates, serve to cut 
off the rarefied air within the receivers when it is desired to leave 
them for any length of time. The cock is also an exchange-cock, 
so as to admit the external air into the receivers. 

Pumps thus constructed have advantages over such as work 
with valves, in that they last longer, exhaust better, and may be 
employed as condensers when suitable receivers are provided, by 
merely reversing the operations of the exchange valve during the 
motion of the piston. 



TABLES. 



TABLE 1. 

THE TENACITIES OF DIFFEKENT SUBSTANCES, AND THE EESISTANCES 
WHICH THEY OPPOSE TO DIRECT COMPEESSION.-See §269. 



SUBSTANCES EXPERIMENTED ON. 



Wrought-iron, in wire from l-20th 
to l-30th of an inch in diame 

ter 

in wire, 1-1 0th of an inch • • 
in bars, Eussian (mean) • • • 
English (mean) • 

hammered 

rolled in sheets, and cut length- 
wise 

ditto, cut crosswise • 
in chains, oval links 6 in. clear, 
iron U in. diameter . . . . 
ditto, Brunton's, with stay across 

link 

Cast Iron, quahty No. 1 . • . . 
2 . . . . 
3* . . . . 

Steel, cast 

cast and tilted 

blistered and hammered • 

shear 

raw 

Damascus 

ditto, once refined .... 
ditto, twice refined .... 

Copper, east 

hammered 

sheet 

wire 

Platinum wire 

Silver, cast 

wire 

Gold, cast 

wire 

Brass, yellow (tine) 

Gun metal (hard) 

Tin, cast 

wire 

Lead, cast 

milled sheet 

wire 



:( 



6o to 91 

36 to 43 

27 

25i 

3o 
14 
18 

2U 



2D 

to 71 
to 8 
to 9I 
44 
60 
59i 

57 
5o 
3i 
36 
44 
8^ 
ID 
21 
27^ 
17 
18 

17 

9 
14 

8 
16 

2 

3 
4-5ths 

U 

hi 



^S 

c a 

S3 2 

£.= 
^1 



Lame 

Telford 
Lame 

Brunei 
Mitis 

Brown 

Barlow 
Hodgkinson 



Mitis 
Eennie 



Mitis 

Eennie 
Kingston 
Guy ton 

Eennie 



Tredgold 
Guy ton 



38 to 41 
37 to 48 
Di to 65 






Hodgkinson 



Eennie 



*The strongest quality of cast iron, is a Scotch iron known as the Devon Hot Blast, No. 3: its tenaci- 
ty is 9J tons per square inch, and its resistance to compression 65 tons. The exiierinients of Major 
Wade on the gun iron at West Point Foundry, and at Boston, give results as high as 10 to 16 tons, and 
on small cast bars, as high as 17 tons.— See Ordnance Manual, 1850, p. 402- 



TABLE I. 
TABLE I — continued. 



429 



SUBSTANCKS rXPKRlMKNTED ON. 



CCS 



zS, 



Stone, slate (Welsh) • • • 

Marble (white) .... 

Givry 

Portland 

Craigleith freestone • • 

Bramley Fall sandstone • 

Cornish granite • . • 

Peterhead ditto • • « 

Limestone (compact blk) • 

Purbeck 

Aberdeen granite • 
Brick, pale red 

red 

Hammersmith (pavior's) • 
ditto (burnt) • • • 

Chalk 

Plaster of Paris 

Glass, plate 

Bone (ox) 

Hemp fibres glued together • 
Strips of paper glued together 
Wood, Box, spec, gravity • 

Ash 

Teak 

Beech 

Oak 

Ditto 

Fir 

Pear 

Mahogany 

Elm 

Pine, American • • • 

Deal, white 



,862 

,6 

j9 

,7 

,77 
,6 

,646 
,637 



5,7 
4 

I 



,o3 

4 

2,2 
41 
i3 



Barlow 



1,4 
1,6 

2;4 

2,7 

2,8 

3,7 

4 

4 

5 

,56 

,8 
I 
1,4 

,22 



,57 
,73 
,86 



Ecnnie 



430 



TABLE II, 



TABLE n. 

OF THE DENSITIES AND VOLUMES OF WATEE AT DIFFEKENT DEGKEES 
OF HEAT, (ACCORDING TO STAMPFEE), FOE EVEEY 2i DEGEEES OF 
FAHEENHEIT'S SCALE.— See § 276. 

(Jakrbuch des Polytechnischen Institutes in fVein, Bd. 16, S. 70). 



t 

Temperature. 


Density. 


Diff. 


V 

Volume. 


Diff. 



32,00 


0,999887 




l,oooii3 




34,25 


0,999950 


63 


i,oooo5o 


63 


36,5o 


0,999988 


38 


1,000012 


38 


38,75 


1,000000 


12 


1,000000 


12 


41,00 


0,999988 


12 


1,000012 


12 


43,25 


0,999952 


35 


1,000047 


35 


45,5o 


0,999894 


58 


1,000106 


59 


47,75 


0,999813 


81 


1,000187 


81 


5o.oo 


0,999711 
0,999587 


102 


1,000289 
1, 000413 


102 


52,25 


124 


124 


54, 5o 


0,999442 


145 


i,ooo558 


145 


56,75 


0,999278 


164 


1,000723 


i65 


59,00 


0,999095 


i83 


1,000906 


i83 


61, 25 


0,998893 


202 


1,001108 


202 


63,5o 


0,998673 


220 


1,001329 


221 


65,75 


0,998435 


238 


1,001567 


238 


68,00 


0,998180 


255 


1,001822 


255 


70,25 


0,997909 


271 


1.002095 


273 


72, 5o 


0,997622 


287 


1,002384 


289 


74,75 


0,997320 


302 


1,002687 


3o3 


77.00 


0,997003 


317 


i,oo3oo5 


3i8 


79,23 


0,996673 


33o 


i,oo3338 


333. 


8i,5o 


0,996329 


344 


i,oo3685 


347 


83,75 


0,995971 


358 


1,004045 


36o 


86,00 


0,990601 


370 


1,004418 


373 


88,25 


0,995219 


382 


1,004804 


386 


90,50 


0,994825 


394 


I,005202 


398 


92,75 


0,994420 


40 5 


i,oo56i2 


410 


95,00 


0,994004 


416 


i,oo6o32 


420 


97,25 


0,993579 


425 


1,006462 


43o 


99,5o 


0,993145 


434 


1,006902 


440 



With this table it is easy to find the specific gravity by means of water at any temperature. 
Suppose, for example, the specific gravity S' in Equation (456), had been found at the tempera- 
ture of 59°, then would Z)// in that equation be 0,999095, and the specific gravity of the body 
referred to water at its greatest density, would be given by 

S = 5' X 0,999095. 



TABLE III. 



431 



TABLE III. 



OF THE SPECIFIC GEAVITIES OF SOME OF THE MOST IMPORTANT BODIES. 
[The density of distilled water is reckoned in this Table at its maxiimim SSJO F. = 1,000]. 



Name of the Body. 



Specific Gravity. 



I. SOLID BODIES. 
(1) Metai^. 



to Lieut. Matzka 



Antimony (of the laboratory) 

Bniss .... 

Bronze for cannon, according 

Ditto, mean • 

Copper, melted 

Ditto, hammered 

Ditto, wire-drawn 

Gold, melted • 

Ditto, hammered 

Iron, wrought 

Ditto, cast, a mean 

Ditto, gray 

Ditto, wliite • 

Ditto for cannon, a mean 

Lead, pure melted 

Ditto, flattened 

Platinum, native 

Ditto, meked • 

Ditto, liammered and wire-drawn 

Quicksilver, at 32° Fahr. 

Silver, pure melted 

Ditto, hammered • 

Steel, cast 

Ditto, wrought 

Ditto, much hardened 

Ditto, slightly 

Tin, chemically pure 

Ditto, hammered 

Ditto, Bohemian and Saxon 

Ditto, English 

Zinc, melted • 

Ditto, rolled • 



(2) Building Stones 



Alabaster • 

Basalt . 

Dolerite • 

Gneiss • 

Granite • 

Hornblende 

Limestone, various kinds 

Piionolite 

Porphyry 

(iunrtz • 

Sandstone, various kinds 

Stones for building 

Syenite • 

Trachyte 

Brick" • 



4,2 


— 


4,7 


7,6 


— 


8,8 


8,4i4 


— 


8,974 


8,708 






7,788 


— 


8,726 


8.878 


— 


8=9 


8,78 






19,238 


— 


19,253 


19,361 


— 


^9'^„ 


7,207 


— 


7,788 


7,25i 






7,2 






7,5 






7,21 


— 


7,3o 


ii,33o3 






11,388 






16,0 





18,94 


20,855 






21,25 






i3,568 


— 


13,598 


10,474 






io,5i 


— 


10,622 


7;9'9 






7,840 






7,818 






7,833 






7,291 






7,299 


— 


7,475 


7,3.2 






7.291 






6,861 


— 


7,2i5 


7,191 




. 


2,7 




3,0 


2,8 


— 


3,1 


2,72 


— 


2.93 


2,5 


— 


2,9 


2,5 


— 


2,66 


2,9 


— 


3.1 


2,64 


— 


2,72 


2,5l 


— 


2,6n 


2,4 


— 


2,6 


2,56 


— 


2.75 


2,2 


— 


2,5 


1,66 


— 


2,62 


2,5 


— 


3, 


2,4 


— 


2.6 


1,41 


— 


1,86 



432 



TABLE III. 



TABLE 111— Continued, 



Name of the Body, 



I. SOLID BODIES. 



(3) Woods. 



Alder . 

Ash 

Aspen • 

Birch . 

Box 

Ehn 

Fir ... 

Hornbeam 

Horse-chestnut 

Larch 

Lime 

Maple 

Oak 

Ditto, another specimen 

Pine, Finns Abies Picea 

Ditto, Pinvs Sylvestris 

Poplar (ItaUan) 

Willow • 

Ditto, white • 



(4) Vaeious Solid Bodies. 

Charcoal, of cork • • • 

Ditto, soft wood 

Ditto, oak 

Coal 

Coke 

Earth, common ..... 

rough sand ..... 

rough earth, with gravel • 

moist sand 

gravelly soil ..... 

clay 

clay or loam, with gravel • 

FHnt, dark 

Ditto, white 

Gunpowder, loosely filled in 

coarse powder . • . • • 

m«sket ditto 

Ditto, slightly shaken down 

musket-powder .... 

Ditto, solid ...... 

Ice .•....•• 

Lime, unslacked ..... 

Eesin, common ..... 

Kock-salt 

Saltpetre, melted 

Ditto, crystallized 

Slate-pencil ...... 

Sulphur ....••• 

Tallow 

Turpentine 

Wax, white ...... 

Ditto, yellow ....•• 

Ditto, shoemaker's 



Specific Gravity. 



Fresh -felled. 

0,8571 
0,9086 
0,7654 
0,9012 
0,9822 
0.9476 
0,8941 
0,9452 
0,8614 
0,9206 
0,8170 
0,9036 
1,0494 
1,0754 
0,8699 
0,9121 
0,7634 
o,7i55 
0,9859 



°'^ 
0,28 

1,573 

1,232 

1,865 



1,92 
2,02 
2,o5 
2,07 

2, ID 
2,48 
2,542 
2,741 

0,886 
0,992 

1,069 
2,248 
0,916 

1,842 
1,089 
2,257 
2,745 
1,900 

1,8 

1,92 

0,942 

0,991 

0,969 

0,965 

0,897 



Dry. 

o,5ooi 
0,6440 
o,43o2 
0,6274 
0,5907 
0,5474 
o,555o 
0,7695 
0,5749 
0,4735 
0,4390 
0,6592 
0,6777 
0,7075 
0,4716 
o,55o2 
0,3931 
0,5289 
0,4873 



0,44 
i,5io 



2,563 
0,9268 



2,24 



TABLE III, 



433 



TABLE lll—Contimied. 



Name of the Body. 


Specific Gravity. 


II. LIQUIDS. 




Acid, acetic ...... 








i,o63 


Ditto, muriatic 














1,211 


Ditto, nitric, concentrated 














1,521 — 1,522 


Ditto, sulphuric, Enirlish 














1.845 


Ditto, concentrated (Nordh.) 














1,860 


Alcohol, free from water 
















0.792 


Ditto, common _ 


















0,824 — 0,79 


Airunoniac, liquid 


















0,875 


Aquafortis, double 


















i,3oo 


Ditto, single • 


















1,200 


Beer 


















I.023 — i,o34 


Ether, acetic • 


















0,866 


Ditto, muriatic 


















0,845 — 0,874 


Ditto, nitric • 


















0,886 


Ditto, sulphuric 


















o,7i5 


Oil, linseed • 


















0,928 — 0,953 


Ditto, olive • 


















0.915 


Ditto, turpentine 


















0,792 — 0,891 


Ditto, whale • 


















0,923 


Quicksilver 


















i3,568 — 13,598 


Water, distilled 


















1,000 


Ditto, rain 


















i,ooi3 


Ditto, sea 


















1,0265 — 1,028 


Wine 


0,992 — i,o38 


III. GASES. 


Water =1. 


Barometer 
30 In. 




Teinp. SSjo F. 


Tern. =3-20 


Atmospheric air = yl-Q := • 








t . 


o,ooi3o 


1,0000 


Carbonic acid gas • 








. 


0,00198 


1,5240 


Carbonic oxide gas • 








• 


0,00126 


0,9569 


Carbureted hydrogen, a maximum 








• 


0,00127 


OQ784 


Ditto, from Coals • 








; 1 


o,ooo3^ 
o,ooo8d 


0.3000 
0,5596 


Chlorine . • • • • 












0,0032I 


2,4700 


Hydriodic gas . • • • 












0,00577 


4,443o 


Hydrogen .... 












0,0000895 


0,0688 


Ilydrosulphuric acid gas • 












o,ooi55 


1,1912 


Muriatic acid gas 












0,00162 


1,2474 


Nitrogen .... 












0,00127 


0,9760 


Oxygen . • • • . 












0,00143 


1,1026 


Phosphurcted hydrogen gas • 












0,001 i3 


0,8700 


Steam at 212° Fahr. 












0,00082 


0,6235 


Sulphurous acid gas • • • • • ... 


0,00292 


2,2470 



28 



434: 



TABLE IV. 



TABLE TV. 

TABLE FOE FINDING ALTITUDES.-See 



284. 



Detached Thermometer. 


tj+t' 


A 


t^ + t' 


A 


t^ + t' 


A 


t^ + t' 


A 


40 


4,7689067 


75 


4,7859208 


no 


4,8022936 


145 


4,8180714 


41 


,7694021 


76 


,7863973 


III 


,8027525 


146 


,8i85i4o 


42 


,7698971 


77 


,7868733 


1 12 


,8032109 


147 


,8189559 


43 


,7703911 


78 


,7873487 


ii3 


,8036687 


148 


,8193975 


44 


,7708801 


79 


,7878236 


114 


,8041261 


149 


,8198387 


45 


,7713785 


80 


,7882979 


.ii5 


,8o4583o 


i5o 


,8202794 


46 


,7718711 


81 


,7887719 


116 


,8o5o395 


i5i 


,8207196 


47 


,7723633 


82 


,7892451 


117 


,8054953 


l52 


,8211594 


48 


,7728548 


83 


,7897180 


118 


,8059509 


1 53 


,8215988 


49 


,7733457 


84 


,7901903 


119 


,8o64o58 


1 54 


,8220377 


5o 


,7738363 


85 


,7906621 


120 


,8068604 


1 55 


,8224761 


5i 


,7743261 


86 


,79n335 


121 


,8073144 


1 56 


,8229141 


52 


,7748153 


87 


,7916042 


122 


,8077680 


i57 


,8233517 


53 


,7753042 


88 


,7920745 


123 


,8082211 


1 58 


,8237888 


54 


,7757925 


89 


,7925441 


124 


,8086737 


159 


,8242256 


55 


,7762802 


90 


,7930135 


125 


,8091258 


160 


,8246618 


56 


,7767674 


91 


,7934822 


126 


,8095776 


161 


,8200976 


57 


,7772540 


92 


,7939504 


127 


,8100287 


162 


,8255331 


58 


,7777400 


93 


,7944182 


128 


,8104795 


1 63 


,8259680 


59 


,7782256 


94 


,7948854 


129 


,8109298 


164 


,8264024 


60 


,7787105 


95 


,7953521 


i3o 


,81,3796 


i65 


,8268365 


61 


)779>949 


96 


,7958184 


i3i 


,8118290 


166 


,8272701 


62 


,7796788 


97 


,7962841 


l32 


,8122778 


167 


,8277034 


63 


,7801622 


98 


,7967493 


i33 


,8127263 


168 


,8281362 


64 


,7806450 


99 


,7972141 


i34 


,8131742 


169 


,8285685 


65 


,7811272 


100 


,7976784 


i35 


,8i362!6 


170 


.8290005 


66 


,7816090 


lOI 


,7981421 


i36 


,8140688 


171 


,8294319 


67 


,7820902 


102 


,7986054 


i37 


,8i45i53 


172 


,8298629 


68 


,7825709 


io3 


,7990681 


i38 


,8149614 


173 


,8302937 


69 


,783o5ii 


104 


,7995303 


139 


,8154070 


174 


,8307238 


70 


,7835306 


io5 


,7999921 


140 


,8i58523 


175 


,83 1 1 536 


71 


,7840098 


106 


,8004533 


141 


,8162970 


176 


,83i583o 


72 


,7844883 


107 


,8009142 


142 


,8167413 


177 


,8320119 


73 


,7849664 


108 


,8013744 


143 


,8171852 


178 


,8324404 


74 


4,7854438 


109 


4,8018343 


144 


4,8176285 


179 


4,8328686 



TABLE IV. 



435 



TABLE lY— continued. 

WITH THE BAROMETER.— See § 284. 



Latitude. 


Attn 


ched Thermometer, 


^ 


B 


T~T' 


C 


c 


0° 


0,0011689 




+ 





3 


,0011624 


0'^ 


0000000 


0,0000000 


6 


,0011433 


I 


0000434 


9,9999066 


9 


,0011117 


2 


0000869 


,9999181 


12 


,0010679 


3 


oooi3o3 


,9998697 


ID 


,0010124 


4 


0001787 


,9998262 


i8 


,0009439 


5 


0002171 


,9997828 


21 


,0008689 


6 


OOO2t)05 


,9997898 


24 


,0007825 


7 


0008089 


,9996909 


27 


,0006874 


8 


0008473 


,9996024 


3o 


,0000848 


9 


0008907 


,9996090 


33 


,0004738 


10 


0004341 


,9990600 


36 


,ooo36i5 


II 


0004775 


,9990220 


39 


,0002433 


12 


0000208 


,9994785 


42 


,0001223 


i3 


0000642 


,9994800 


45 


,0000000 


14 


0006076 


,9998916 


48 


9,9998775 


10 


0006010 


,9993481 


49 


,9998872 


16 


0006048 


,9998046 


5o 


,99979^^7 


17 


0007877 


,9992611 


5i 


,9997066 


18 


0007810 


,9992176 


52 


,9997167 


19 


0008244 


,9991741 


53 


,9996772 


20 


0008677 


,9991805 


54 


,999638r 


21 


00091 1 1 


,9990870 


55 


,9990990 


22 


0009044 


,9990435 


56 


,9990613 


23 


0009977 


,9990000 


il 


,99902,37 


24 


001041 I 


,99.S9564 


58 


,9994866 


25 


00 1 0844 


,99-9129 


59 


,9994502 


26 


0011277 


,99«»694 


6o 


,9994144 


27 


0011710 


,9988208 


63 


,9993110 


28 


00 1 2 1 43 


,9987823 


66 


,9992161 


29 


0012076 


,9987887 


69 


,9991293 


3o 


U0l300Q 


,99.S6902 


75 


,99^9802 


3i 


0018442 


9,9986016 


81 


,9980854 








90 


9,9988300 









436 



TABLE V. 



TABLE Y. 

COEFFICIENT VALUES, FOR THE DISCHAEGE OF FLUIDS THEOUGH THIN 
PLATES, THE OEIFICES BEING EEMOTE FKOM THE LATEEAL FACES 
OF THE VESSEL.— See § 300. 





Values 


of the coefficients for orifices whose smallest dimensions or 


Head of fluid 






diameters are — 




above the 
centre of the 
























orifice, in feet. 


ft. 


ft. 


ft- 


ft. 


ft. 


ft- 




0,66 


0,33 


0,16 


0,08 


0,07 


o,o3 


o,o5 












0,700 


0,07 








0,627 


0,660 


0,696 


o,i3 






0,618 


0,632 


0,637 


0,685 


0,20 




0,592 


0,620 


0,640 


0,656 


0,677 


0,26 




0,602 


0,625 


0,638 


0,655 


0,672 


o;33 


0,593 


0,608 


o,63o 


0,637 


0,655 


0,667 


0,66 


0,596 


o,6i3 


o,63i 


0,634 


0,654 


0,655 


1,00 


0,601 


0,617 


o,63o 


0,632 


0,644 


o,65o 


1,64 


0,602 


0,617 


0,628 


o,63o 


0,640 


0,644 


3,28 


o,6o5 


o,6i5 


0,626 


0,628 


0,633 


0,632 


5,00 


o',6o3 


0,612 


0,620 


0,620 


0,621 


0,618 


6,65 


0,602 


0,610 


o,6i5 


o,6i5 


0,610 


0,610 


32,75 


0,600 


0,600 


0,600 


0,600 


0,600 


0,600 



In the instance of gas, the generating head is always greater than 6,65 ft., and the coefficient 0,6, 
or 0,61, is taken in all cases. 

For orifices larger than 0,66 ft., the coefficients are taken as for this dimension ; for orifices smaller 
than 0,03 ft., the coefficients are the same as for this latter ; finally, for orifices between those of the 
table, we lake coefficients whose values are a mean between the latter, corresponding to the given head. 



TABLE VI. 



437 



TABLE YI. 

EXPERIMENTS ON FRICTION, WITHOUT UNGUENTS. BY M. MORIN. 

Tlie surfaces of friction were varied from o,o3336 to 2,7987 square feet, the pressures from 
88 lbs. to 22o5 lbs., and the velocities from a scarcely perceptible motion to 9,84 feet per 
second. The surfaces of wood were planed, and those of metal filed and polished with the 
greatest- care, and carefully wiped after every experiment. The presence of unguents was 
especially guarded against. — See § 808. 





Friction of 


Friction 


TF 


SURFACES OF CONTACT. 


Motion.* 




aUIE^ 


CENCE.f 


^ s 




<u 


♦^ d 









£■§ 


.I'se 


S.2 


|-5 1 




'i'-X 




-^ 


U 








5'= 


— CD 


d'o 


^ 


~ 


Oak upon oak, the direction of the fibres ) 
being parallel to the motion • • • | 


0,478 


25= 


33' 


0,625 


32«> I' 


Oak upon oak, the directions of the fibres "] 














of tiie moving surface being perpen- 
dicular to those of the quiescent sur- 


0,324 


17 


58 


0,540 


28 


23 


face and to the direction of the motion^ J 
Oak upon oak, the fibres of the both sur- 


























faces being perpendicular to the direc- I 


0,336 


18 


35 








tion of the motion 














Oak upon oak, the fibres of the moving^ 
surface being perpendicular to tlie snr- 1 






■ 




















face of contact, and those of the surface I 


0,192 


10 


52 


0,271 


i5 


10 


at rest parallel to the direction of the 














TnntW^ti ••■■•«•... 














Oak upon oak, the fibres of both surfaces , 














being perpendicular to the surface of I 




• 




0,43 


23 


17 


contact, or the pieces end to end • • ) 














Elm upon oak, the direction of the fibres I 
being parallel to the motion • ' • ) 


0,432 


23 


21 


0,694 


34 


46 


Oak upon elm, ditto§ 


0,246 


i3 


5c 


0,376 


20 


37 


Elm upon oak, the fibi'es of the moving"] 














surface (the elm) being perpendicular to [ 
those of the quiescent sui'face (the oak) ( 


o,45o 


24 


16 


0,570 


29 


41 


and to the direction of the motion- • J 














Ash upon oak, the fibres of both surfaces 














being parallel to the direction of the > 


0,400 


21 


49 


0,570 


29 


41 


motion 














Fir upon oak, the fibres of both surfaces 














being parallel to the direction of the > 


0,355 


19 


33 


0,520 


27 


29 


motion 














Beach upon oak, ditto 


o,36o 


'9 


48 


0,53 


27 


56 


Wild pear-tree npon oak, ditto • 


0,370 


20 


19 


0.440 


23 


45 


Service-tree upon oak, ditto .... 


0.400 


21 


49 


0.570 


21 


4' 


Wrought iron ujion oak, dittoll • 


0,619 


3i 


47 


0,619 


3i 


47 



* Tho friction in this case varies but very .slinlitly from tho iiicrin. 

t 'I'lii- Criclion in this ciise varies consi(lerai)ly from tlie mean. In all the experiments the surfaces 
had been 1') miniue-< in contact. 

t The (lini<n>ioiis r)f the surfaces of contact were in this experiment .947 square feet, nn<l the results 
were nearly iiniforin. When the dimensions were diminished to ,fl4H a learinp of liie filire became appa- 
rent ill tli<;'c:isf; of motion, and there were sym|it.ims of the coinlnistioii of the wood ; from llie.se cir- 
cumst.inces there resulted an irretrularity in iIm; friction indicative of e.vccs.-iv(! |)resMire. 

<^ It is worthy of rem^irk that the friction of oak upon elm is but five-ninths of ih^-t of elm upon oak, 

II [n the experiments in wtiich one of the surfaces was of nietil, small particles of the nif;ial liei^au, 
after a time, to he apparent upon the wood, pivin}: it a polished metallic appearance ; these were at every 
experiineiU wiped ofi'; they indicated n wearing' of the itielal. 'I'he frictmn of motion and Ihatof quios- 
cence, in these ex|»eriment3, coincided. The results were remarkably uniforui. 



438 



TABLE VI. 



TABLE Yl— continued. 





Friction of 


Friction of 


SURFACES OF COxNTACT. 


Motion. 




Q.UIESCKNCE. 






CD 









c S 




„ 


c 


V =-- " 




QJ-3 


fee 




©•^ 


" = = 




is 


1 




1) -< 






O o 


l<^ 


6-= 




Wrought iron upon oak, the surfaces ) 
being greased and well wetted- • • ) 


0,256 


14° 22' 


0,649 


33° 0' 


Wrouglit iron upon elm 


0,232 


14 


9 


. 


• 


Wrouglit iron upon cast iron, the fibres ) 
of the iron behig parallel to the motion \ 


0,194 


10 


59 


0,194 


10 59 


Wrouicht iron upon wrought iron, the ) 












fibres of both surfaces being parallel - 


o,i38 


7 


52 


0,137 


7 49 


lo the motion ) 












Cast iron upon oak, ditto 


0,490 


26 


7 






Ditto, the surfaces being greased and \ 
wetted ) 








0,646 


32 52 


Cast iron upon elm 


0,195 


II 


3 






Cast iron upon cast iron 


0,132 


8. 


39 


0,162 


9 i3 


Ditto, water being interposed between ) 
the surfaces j 


0J14 


-17 


26 






Cast iron upon brass 


0,147 


8 


22 






Oak upon cast iron, the fibres of the wood i 












being perpendicular to the direction \ 


0,372 


20 


25 






of the motion ) 












Hornbeam upon cast iron— fibres paral- / 
lei to motion (" 


0,394 




3i 






21 






Wild pear-tree upon cast iron— fibres \ 
parallel to the motion ] 


o,436 


23 


34 










Steel upon cast iron 


0,202 


II 


26 






Steel upon brass 


0,1 52 


8 


39 






Yellow Clipper upon cast iron . • • • 


0.189 


10 


49 






Ditto oak .... 


0,617 


3i 


41 


0,617 


3i 41 


Brass upon cast iron 


0.217 


12 


i5 






Brass upon wrought iron, the fibres of) 
the iron being parallel to the motion • j 


0,161 










9 


9 






Wrought iron upon brass 


0.172 


9 


46 






Brass upon brass 


0,201 


11 


22 






Black leather (curried) upon oak* • 


0.265 


14 


5i 


0,74 


36 3i 


Ox hide (such as that used for soles and 












for the stuffing of pistons) upon oak, V 


0,52 


27 


29 


o,6o5 


3i II 


rough 












Ditto ditto ditto smooth • 


0,335 


18 


3i 


0,43 


23 17 


Leather as above, polished and hardened ) 
by hammering \ 


0.296 


16 


3o 






Hempen girth, or pulley-band, fsangle") 












de chanvre,) u^ion oak, the fibres of! 
the wood and tiie direction of the cord [ 


0,52 


27 


29 


0,64 


32 38 


being parallel to the motion • • -J 












Hempen matting, woven with small ( 
cords, ditto. ^ ) 


0,32 


n 


45 


o,5o 


26 34 


Old cordage, li inch in diameter, dittof 


0,52 


27 


29 


0,79 


38 19 



* The friction of motion was very nearly the same whether the surface of contact was the inside 
or the oiitsiile of the j-kin. — The constavcy of the coefficient of tlie friction of motion was equally ap- 
parent in the rough and the smooth skins. 

t All the above experiments, e.xcept that wilh curried black leather, presented the phenomenon of 
a change in the polish of the surfaces of friction — a state of their surfaces necessary to, and dependent 
upon, their motion upon one another. 



TABLE VI. 

TABLE Yl— continued. 



439 





Friction of 


Friction of 


SURFACES OF CONTACT. 


Motion. 




auiE 


SCKNCE. 


« d 




6 


■" s 


. 




0) -r: 


pi 


= 


~ ~ 7: 






1" 


-.2 


ll 


Si: .2 


• 


a-o 


j^tf 







Calcareous oolitic stone, used in building, ^ 
of a moderately hard quality, called ! 
stone of Jaumout— upon the same | 












0,64 


32° 


38' 


0,74 


36° 3i' 














oLone ..........J 

Hard calcareous stone of Brouck, of a1 












light gray color, susceptible of taking 1 
a tine polish, (the mu^chelkalk,) mov- [ 


0,38 


20 


49 


0,70 


35 


iug upon the same stone j 












The soft stone mentioned above, upon ^ 
the hard ) 


0,65 


33 


2 


0.75 


36 53 


The hard stone mentioned above upon 
the soft 


0,67 


33 


5o 


0,75 


36 53 


Common brick upon the stone of Jaumont 


0,65 


33 


2 


0,65 


33 2 


Oak upon ditto, the fibres of the wood 












being perpendicular to the surface of > 


0,38 


20 


49 


0,63 


32 i3 


the stone ) 












Wrought iron upon ditto, ditto • 


0,69 


34 


37 


0,49 


26 7 


Common bi-ick upon the stone of Brouck 


0,60 


3o 


58 


0,67 


33 5o 


Oak as before (endwise) upon ditto • 


0.38 


20 


49 


0,64 


32 38 


Iron, ditto ditto • • 


0,24 


i3 


3o 


0,42 


22 47 



uo 



TABLE VII. 



TABLE YII. 

EXPEEIMENTS ON THE FKICTION OF UNCTUOUS SUEFACES. 
BY M. MOKIN.— See § 308. 

In these experiments the surfaces, after having been smeared with an unguent, were 
wiped, so that no interposing Layer of the unguent prevented their intimate contact. 





Friction of 


Friction op 


SURFACES OF CONTACT. 


Motion. 




auiE 


SCENCE. 






6 









= o 




c s 






<u-z: 


fct-^ 




.5^ - 


ts.'^ a 




!•= 




'"■H 


•1 «| 




*gh 


"i 5'"'" 


1'- 


"s "sc'm 




c -, 




CD 


c-_ 


.- = Qi 


* 


O o 


hJ< 


Si 


c r 


^<t^ 


Oak upon oak, the fibres being parallel to ) 
the motion • • • • • ) 


o,io8 


6° 


10' 


0,390 


21° 19' 


Ditto, the fibres of the moving body be- ( 


0,143 


8 




o,3i4 


17 26 


ing perpendicular to the motion- 


[ 


9 


Oak upon elm, fibres partillel' 




o,i36 


7 


45 






Elm upon oak, ditto 




0,1 19 


6 


48 


0,420 


22 47 


Beech upon oak, ditto • 




o,33o 


j8 


16 






Elm upon elm, ditto 




0,140 


7 


59 






Wrought iron upon elm, ditto 




o,i38 


7 


52 






Ditto upon wrought iron, ditto 




0,177 


10 


3 






Ditto upon cast iron, ditto • • . 






. 




0,118 


6 44 


Cast iron upon wrought iron, ditto 




0,143 


8 


9 






Wrought iron upon brass, ditto • 




0,160 


9 


6 






Brass upon wrought iron 




0,166 


9 


26 






Cast iron upon oak, ditto 




0,107 


6 


7 


0,100 


5 43 


Ditto upon elm, ditto, the unguent beiuc 
tallow ..... 


1 


0,125 


7 


8 






Ditto, ditto, the unguent being hog't 
lard and black lead • 


1 


o,i37 


7 


49 






Elm upon cast iron, fibres parallel • 




o,i35 


7 


42 


0,098 


5 36 


Ca.<t iron upon cast iron 




0,144 


8 


12 






Ditto upon brass .... 




0,l32 


7 


32 






Brass upon cast iron • 




0,107 


6 


7 






Ditto upon brass .... 




o,i34 


7 


38 


0,164 


9 19 


Copper upon oak .... 




0,100 


5 


43 






Yellow copper upon cast iron 




o,ii5 


6 


34 






Leather (ox liide) well tanned upon cas 
iron, wetted . . . • 


^ 1 


0,229 


12 


54 


0,267 


14 57 


Ditto upon brass, wetted • • 


0,244 


i3 


43 







TABLE VIII, 

TABLE Till. 



441 



EXPEEIMENTS ON FRICTION WITH UNGUENTS INTEEPOSED. BY M. MOEIN. 

The extent of the surfaces in these experiments bore such a relation to the pressure, as 
to cause them to be separated from one another throughout by an interposed stratum of 
the unguent.— See § 308. 





Friction 


Friction 






OF 


OF 




SURFACES OF CONTACT. 


Motion. 


Quiescence. 


UXGUENTS. 


^ 






_0 2 


1^1 






e = u 

1 - 


a 2 




Uak upon oak, fibres parallel • 


0,164 


0,440 


Dry soap. 


Ditto ditto 


0,075 


0,164 


Tallow. 


Ditto ditto 


0,067 




Hog-s lard. 


Ditto,, fibres perpendicular • 


o;o83 


0,254 


Tallow. 


Ditto ditto 


0,072 




Hog's lard. 


Ditto ditto 


0.230 


. 


"Water. 


Ditto upon elm, fibres parallel 


o.i36 




Drv soap. 


Ditto ditto 


0,073 


0,178 


Taflow. 


Ditto ditto 


0,066 




Hoff-s lard. 


Ditto upon cast iron, ditto 


0,080 




Tallow. 


Ditto upon wi ought iron, ditto 


0.098 




Tallow. 


Beech upon oak, ditto 


o.o55 




Tallow. 


Elm upon oak, ditto • 


0.137 


0,411 


Drv soap. 


Ditto ditto 


0.070 


0,142 


Tallow. 


Ditto ditto 


0.060 




IlogV lard. 


Ditto upon elm, ditto • 


0,139 


0,217 


Di-v soap. 


Ditto upon cast iron, ditto 


0,066 


• 


Tailow. 

Greased, and 


"Wrought iron upon oak, ditto • 


0,256 


0,649 


■ sat nrated with 
water. 


Ditto ditto ditto • 


0,214 


. 


Drv'Voap. 


Ditto ditto ditto • 


o,o85 


0,108 


Tallow. 


Ditto upon elm, ditto • 


0,078 




Tallow. 


Ditto ditto ditto • 


0,076 


. 


Hotr's lard. 


Ditto ditto ditto • 


0.055 


. 


Olive oil. 


Ditto upon cast iron, ditto 


o,io3 




Tallow. 


Ditto ditto ditto • 


0,076 




Hog's lard. 


Ditto ditto ditto • 


0,066 


0,100 


Olive oil. 


Ditto upon wrought iron, ditto 


0,082 




Tallow. 


Ditto ditto ditto • 


0,081 




Hog's lard. 


Ditto ditto ditto • 


0,070 


o,ii5 


Olive oil. 


Wrought iron upon brass, fibres ) 
parallel • • • • • j 
Ditto ditto ditto • 


o,io3 


. . 


Tallow. 


0,075 




Hosr's lard. 


Ditto ditto ditto • 


0,078 


. 


Olive oil. 


Cast iron upon oak, ditto • 


0,189 


• 


Dry soap. 
Greased, and 


Ditto ditto ditto • 


0,218 


0,646 


• saturated with 
water. 


Ditto ditto ditto • 


0,078 


0,100 


Tallow. 


Ditto ditto ditto • 


0,075 




Hog's lard. 


Ditto ditto ditto • 


0,075 


0,100 


Olive oil. 


Ditto upon elm, ditto • 


0.077 




Tallow. 


Ditto ditto ditto • 


o,o6i 




Olive oil. 


Ditto ditto ditto • 


0,091 




j liog's lard and 
1 jiluiiibiigo. 


Ditto, ditto upon wrought iron 


. 


0,100 


Tallow. 


Cast iron upon cast iron • 


o,3i4 


. 


Water. 


Ditto ditto 


0,197 


• 


Soap. 



4:4:2 



TABLE VIII. 



TABLE YllL— continued. 





Friction 


Friction 






OF 


OF 




SURFACES OF CONTACT. 


Motion. 


Q.UIESCENCK. 


UNGUENTS. 


^ 


^ 




1 § 


1 § 






£C^ 


e = o 






8 e 


£ 




Cast iron upon cast iron • 


0,100 


0,100 


Tallow. 


Ditto ditto 




0,070 


0,100 


Hogs' lard. 


Ditto ditto 




0,064 




Olive oil. 


Ditto ditto 




o,o55 


. 


j Lard and 
j plumbago. 
Tallow. 


Ditto upon brass • 




o,io3 


, 


Ditto ditto 




0,075 




Hogs' lard. 


Ditto ditto 




0,078 


. 


Olive oil. 


Copper upon oak, fibres paralle 
Yellow copper upon cast iron 




0,069 


0,100 


Tallow. 




0,072 


o,io3 


Tallow. 


Ditto ditto 




0,068 


. 


Hogs' lard. 


Ditto ditto 




0.066 


. 


Olive oil. 


Brass upon cast iron- 




0,086 


0,106 


Tallow. 


Ditto ditto 




0,077 




Olive oil. 


Ditto upon wrought iron 




0,081 


. 


Tallow. 


Ditto ditto 






0,089 


• • 


Lard and 
plumbago. 


Ditto ditto 






0,072 


. 


Olive oil. 


Ditto upon brass 






o,o58 


. 


Olive oil. 


Steel upon cast iron 






o,io5 


0,108 


Tallow. 


Ditto ditto 






0.081 




Hogs' lard. 


Ditto ditto 






0,079 


. 


Olive oil. 


Ditto upon wrough 


t iron 




0,093 


. 


Tallow. 


Ditto ditto 






0,076 


. 


Hogs' lard. 
Tallow. 


Ditto uppn brass 






o,o56 


. 


Ditto ditto 


. » 




o,o53 


. 


Olive oil. 


Ditto ditto 






0,067 


' . . 


Lard and 
plumbago. 








Greased, and 


Tanned ox hide upon cast iron 


0,365 


• 


- saturated with 
water. 


Ditto ditto 


0,1 59 


. 


Tallow. 


Ditto ditto 




o;i33 


0,122 


Olive oil. 


Ditto upon brass • 




0,241 




Tallow. 


Ditto ditto 




0,191 


. 


Olive oil. 


Ditto upon oak, • 




0,29 


0,70 


Water. 


Hempen fibres not twisted, niov-' 








inp: upon oak, the fibres of the 






Greased, and 


hemp being placed in a direc- 


0,332 


0,869 


J saturated with 


tion perpendicular to the direc- f 




water. 


tioii of the motion, and those j 








of the oak parallel to it • • j 

The same as above, moving upon (^ 

Ciist iron • • • • ) 








0,194 


• • 


Tallow. 


Ditto 


o,i53 




Olive oil. 


Soft calcareous stone of Jaumont' 
upon the same, with a layer of 














mortar, of sand, and lime inter- ► 




0)74 




posed, after from 10 to 15 min- 








utes' cortact. 









TABLE IX, 



^3 



TABLE IX. 

FEICTIOX OF TEU^^NIONS IN THEIE BOXES.— See S 314. 



KLNDS OF MATERIALS. 



Trunnions of cast iron and 
boxes of cast iron. 



Trunnions of cast iron and 
boxes of brass. 



Trunnions of cast iron and 
boxes of ligDum-vitae. 



Trunnions of wrought iron 
and boxes of cast iron. 



Trunnions of wrouslit iron 
and boxes of brass. 



Trunnions of wrought iron 
and boxes of lignuni-vi- 
t£)e. 

Trnn'nions of brass and ( 
boxes of brass. ) 

Trunnions of brass and | 
boxes of cast iron. \ 

Trunnions of liirnum-vitae ( 
and boxes of cast iron, j 

Trunnions of lignum-vita? ) 
and boxes of lignum- > 
vitai. ) 



STATE OF SURFACES. 



Unguents of olive oil, hogs' lard, 
and tallow .... 

The same unguents moistened witL 
water ..... 

Unguent of asphaltum 

Uueriious ..... 

Unctuous and moistened with wa- 
ter 

Unguents of olive oil, bogs' lard, 
and tallow . . ^. . 

UnctuMus ..... 

Unctuous and moistened with wa- 
ter 

Very slightly unctuous 

Witiiout unguents • 

Ung-uents of olive oil and liosfs' ( 
lard . . . ■ " • ) 

Unctuous with oil and hogs' lard 

Unciuous with a inixtureof hogs' 
lard and plumbago 

Unguents of olive oil, tallow, anJ 
hugs- lard .... 

Unguents of olive oil, hogs' lard, 
and tallow 

Old uniruents Imrdened • 
Unciuous and moistened with wa- 
ter 

Very slightly unctuous 

Unguents of oil or hogs' lard . 
Unctuous . ." .~ . 

Unguent of oil- 
Ungueiit of liogs' lard 

Unguents of tallow or of olive oil. 

Unguents of hogs' lard 
Unctuous- .... 

Unfjuent of hocrs' lard 



Ratio of friction to 
pressure when the 
unfrueiit is renewed. 



By the 

ordiiiiiry 
method. 



0.07 

to 

o,oS 

0.08 

o.o54 

0,14 

o.U 

( 0,07 
■ to 

( 0.08 

0,16 

0.16 
0,19 
o.i8 



Or. con- 

liiaiuuviv, 



0.14 




f 0-07 1 




\ to \ 


o,o54 


I 0,08 J 




( 007 1 




1 08 ) 


0.034 




0.09 


• 


0.19 


. 


0.2J 




0,11 


. 


0.19 


• 


O.IO 




0,09 






0.045 




o,o52 


0,12 


. 


0,13 


• • 



0,0D4 



0.034 

0,034 



o.o54 



0,090 



0,07 



444: 



TABLE X. 



TABLE X. 

OF WEIGHTS NECESSARY TO BEND DIFFERENT ROPES AROUND A WHEEL 

ONE FOOT IN DIAMETER.— See § 310. 



No. 1. White Ropes — new and dky. 
Stiffnems proportional to the square of the diameter. 



Diameter of rope 
in incli.s. 


Natural .stiffness, 
or value of" K. 


Stiffness for load of 
1 lb., or value of /. 


0,39 
0-79 

3,13 


lbs. 
0,4024 

1,6097 

6,43^9 

25,7553 


lbs. 

0.0079877 
o,o3i95oi 
0,1276019 
o,5i 12019 



No. 2. White Ropes — new and moi.^tened with 

WATER. 

Stiffness proportional to square of diameter. 



Diameter of rope 
ia inches. 


Natural stiffness, 
01- value of ^. 


Stiffness for load ol 
1 lli., or value of /. 


0,39 

0:79 

1,57 
3,i5 


lbs. 

0,8048 

3,2194 

12.8772 

5i.5iii 


lbs. 

0,0079877 
o.o3i95oi 
0.1278019 
0,5112019 



Squires of the ratios 


of diameter, or val- 


ues of d?. 


Ratios d . 


Squ res 


1. 00 


1,00 


1. 10 


1,21 


1.20 


1,44 


i,3o 


1,69 


1,40 


1,96 


1.30 


2.25 


1,60 


2.56 


1.70 


2.89 


1,80 


3.24 


1.90 


3.61 


2.00 


4;00 



No. 3. White Ropes — half worn and dry. 

Stiffness proportional to the square root of the cube of 
the diu7neter. 



Diameter of rope 
in inches. 


Natural Stiffness, 
or value of K. 


Sllft'ness for load of 
i 11)., or value of/. 


0.39 
0.79 

1.57 
3,i5 


lbs. 
0,40243 
i,i38oi 
3,21844 
9,ioi5o 


0.0079877 
0.0525889 
0.0638794 
0,1806573 



No. 4. White Ropes — half worn and moistened 
with water. 

Stffnessp)roportiomd to the square root of the cule of 
the dlanu'ter. 



Diameter of rope 
in inches. 


Natural Stiffness, 
or value of K. 


Stilfie^s for lo;ui of 
1 11)., or value of 7. 


0,39 
0.79 
1,57 
3,i5 


lbs. 

0,8048 

2.2761 

6.4324 

i8,2o37 


lbs. 

0,0079877 

0,0525889 

0,0638794 

0,1806573 



Square roots of the 


cubes of the ritios 


ot diameter, or val- 


. ,3 


lies of (/-2- 


Ratios or 


Power ? 


d. 






"'• fi?2- 


1. 00 


1,000 


1. 10 


I.I54 


1.20 


i,3i5 


i.3o 


1.482 


1.40 


1.657 


1.30 


1.837 


1,60 


2.024 


1. 10 


2.217 


I. So 


2 413 


1.90 


2,619 


2,00 


2.828 



APPEIs^DIX. 445 

TABLE X — continued. 

No. 5. Tarred Eopes. 

Stiffness proportional to tlie number of yarns. 

[These ropes are usually made of three strands twisted around each other, each strand being com- 
3d of a certain number of yarns, also twisted about each other in the same manner.] 



No. of yarns. 


Weight of 1 font in 
length of rojje. 


Naturnl siiffness, or 
vtt lue of K. 


Stiffness for load of 
1 lb., or value of /. 


6 
i5 
3o 


lbs. 
0,0211 

0,0497 
l,oi37 


lbs. 
0,1 534 

0,7664 
2,5297 


lbs. 
0,0080198 
0,0198796 
0,0411799 



APPENDIX. 

No. I. 

Take the usual formulas for the transformation of co-ordinates from 
one system to another, both being rectangular, viz : 

X ^= a x' -\- b I/' -\- c z' , 1 

y = a'x' +b'i/ -{-c'z', J. (1) 

z^a"x' -\-b"y' + c"z'; J 

in which a, b, &c., denote the cosines of the angles which the axes of 
the same name as the co-ordinates into which they are respectively 
multiplied make with the axis of the variable in the first member. 
And hence, 

x^ = ax -i- a' 7/ -^ a'^ z, ' 

y' = bx + b' y + b" z, { (2) 

z' — ex -{- c' y -\- c" z ; 

Multiply the first of (2) by b, the second by a, and tako the dif- 
ference of the products ; we get 

bx' - ay' -y{a'b-ab') + z{a''b -ab"); • • • (3) 
again, multiply the first by c, the third by a, and take the dilierence 
of products ; we have 

ex' — az' — y{aJ c — a c') + z {a" c — a c") • • (4) 
Find the value of y in (4), substitute in (3), and reduce, we find 
Az^ijj c' - b' c)x' + {a' c ^a c') y' + (a 6' -^ a' b) z', 



446 APPENDIX. 

in which 

A= c{a' b" - a" h')^-c' {a" h - ah") + c" {ah' - a' h) , 
dividing by A, and subtracting the result from the third of Eqs. (1) 
we have 
/ ,, hc' — h'c\ , , /-,, a'c — ac'\ ,, / ,, ah'—a'h\ , 

and since x' , y' and z' are wholly arbitrary, w^e have 



bc'-b'c ^ ^„ 
a' : = 0; ^>" 



a'c — ac' ^ ,, ah' — a'h ^ ,^. 
= 0; c" -, = ; . (5) 



A ' A ' A 

transposing, clearing the fraction, squaring, adding, collecting the co- 
efficients of c'^, 5 '2, a'2, and reducing by the relations 

a2 + ^2 _|. ^,2 ^ 1 . ^'2 ^ y2 _|_ c'2 ^ 1 . 

a2 + Z*2 ^ 1 - C2 ; C2 + Z»2 ^ 1 — a2; «2 _|_ c2 = 1 _ b2, 

there will result 

^2= 1 _(^aa' + hV ^ccj. 
But 

aa' + Z-^' + cd = 0, 

whence A =: 1, and, Eqs. (5), 

a" —h d — h' c\ h" — a' c — ac' ] c" — ah' — a' h. 

No. II. 

To find the radius of curvature of any curve, and its inclination 
to the co-ordinate axes. 

Take the centre of curvature as the centre of a sphere of which 
the radius is unity. Through the same 
point draAv the line O X, parallel to the 
axis X, and another O T, parallel to the 
tangent to the arc M N^ of osculation. K^--^ 

The planes of these lines and of the ra- 
dius of curvature will cut from the sphere / 
the spherical triangle A B C, oi which the ^" 

side B C is 90°, AC the angle which the radius of curvature makes 
with the axis x, and A B the angle which the tangent to the curve 
makes with the same axis. Make 

^ =z O R z=z radius of curvature, 
^'=AC;c=:AB; C = A C B. 



\ 



.4 



-X 



APPENDIX. 44:7 

Then will 

dx . .. ^ 

cos c = -7- = sm d . cos G ; 
d s 

differentiating, and regarding C as constant, 

dx 
d —- — cos &'. d6\ cos C ; 
d s 

but d&'.cosC is the projection of the arc d 6' on the osculatory 

plane, whence 

d&'. cos C =z — . 

p 

Substituting this above, we find 

d X 

cos r — p • ; 

d s 

and denoting by &" and &'", the angles which the radius makes 
with the axes y and z, respectively, we may write 

^dx dy dz 

d— . d-f- d~ 

cos ^' = p r- ; cos^" = p — : cos^'"=:p — •• •(!) 

^ d s ^ d s d s ^ 

Squaring, adding and reducing by the relation. 

cos2 6' + cos2 6" + cos2 d'" =2 1, 

we have 

d s 



v/(4,)"+("^-y+«f;)" 

performing the operations indicated under the radical sign, and redu- 
cing by the relations 

ds^ =: dx"^ -{- dif + d ^2, 

d^ s d s = d'^ X d X -{- d'^ y d y + d"^ z d z, 
we find 

_ ds' ^ 

^ """ ^/{d?xf + {d^ yf Jr{d:'zf~- (d' s}^ ' ' * * * (^) 
If s be taken as the independent variable, then will d'^ 5 = 0, and 
Eqs. (1) and (2) become 

cos.'=p.;Jlf;cosr = p.;ilf;cosr'=p.J|;. • (3) 

d .S-2 

(A) 



■^(d' xf + (d' yf + {d' zf ' 



448 APPENDIX. 

No. I I I 

To integrate the partial differential equation 

transpose and divide by i>, and we have 
dq p dq 

and because g' is a function of p and jD, we have 

and substituting the value of -7-yri 

d a D'dp — y'P'dD 

1-1 

multiplying and dividing hjy-D-py , 

,1-1 1 

^ D'--p7 'dp-pJ'dB 
dq y-D 7 



but 

D 


dp 

1 
'7* 


1 J ' Z>2 

1-1 1 

p7 .dp--p7-dD - 


and making 




1 

rfp- 1 --^ Vi);' 


we may write 




p7 

1 1 



and by integrating 

in which F^ denotes any arbitrary function. 



